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# 2.3e: Exercises - Transformations

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### A: Concepts

Exercise $$\PageIndex{A}$$

1) When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?
2) When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?
3) When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?
4) When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the $$x$$-axis from a reflection with respect to the $$y$$-axis?
5) How can you determine whether a function is odd or even from the formula of the function?

1. A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.

3. A horizontal compression results when a constant greater than $$1$$ is multiplied by the input. A vertical compression results when a constant between $$0$$ and $$1$$ is multiplied by the output.

5. For a function $$f$$, substitute (−x) for (x) in $$f(x)$$. Simplify. If the resulting function is the same as the original function, $$f(−x)=f(x)$$, then the function is even. If the resulting function is the opposite of the original function, $$f(−x)=−f(x)$$, then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

### B: Describe transformations of a function written in function notation

Exercise $$\PageIndex{B}$$

$$\bigstar$$ Describe how the graph of the function is a transformation of the graph of the original function $$f$$.

 6) $$y=f(x−49)$$ 7) $$y=f(x+43)$$ 8) $$y=f(x+3)$$ 9) $$y=f(x-4)$$ 10) $$y=f(x)+5$$ 11) $$y=f(x)+8$$ 12) $$y=f(x)−2$$ 13) $$y=f(x)−7$$ 14) $$y=f(x−2)+3$$ 15) $$y=f(x+4)-1$$ 16) $$g(x)=f(−x)$$ 17) $$g(x)=−f(x)$$ 18) $$g(x)=6f(x)$$ 19) $$g(x)=4f(x)$$ 20) $$g(x)=f(2x)$$ 21) $$g(x)=f(5x)$$ 22) $$g(x)=f \left(\tfrac{1}{5}x \right)$$ 23) $$g(x)=f \left(\tfrac{1}{3}x \right)$$ 24) $$g(x)=−f(3x)$$ 25) $$g(x)=3f(−x)$$

7. The graph of $$f(x+43)$$ is a horizontal shift to the left $$43$$ units of the graph of $$f$$.

9. The graph of $$f(x-4)$$ is a horizontal shift to the right $$4$$ units of the graph of $$f$$.

11. The graph of $$f(x)+8$$ is a vertical shift up $$8$$ units of the graph of $$f$$.

13. The graph of $$f(x)−7$$ is a vertical shift down $$7$$ units of the graph of $$f$$.

15. The graph of $$f(x+4)−1$$ is a horizontal shift left $$4$$ units and vertical shift down $$1$$ unit of the graph of $$f$$.

17. The graph of $$g$$ is a vertical reflection (across the x-axis) of the graph of $$f$$.

19. The graph of $$g$$ is a vertical stretch by a factor of 4 of the graph of $$f$$.

21. The graph of $$g$$ is a horizontal compression by a factor of $$\frac{1}{5}$$ of the graph of $$f$$.

23. The graph of $$g$$ is a horizontal stretch by a factor of 3 of the graph of $$f$$.

25. The graph of $$g$$ is a horizontal reflection across the y-axis and a vertical stretch by a factor of 3 of the graph of $$f$$.

### C: Graph transformations of a basic function

Exercise $$\PageIndex{C}$$

$$\bigstar$$ Begin by graphing the basic quadratic function $$f(x)=x^2$$. State the transformations needed to apply to $$f$$ to graph the function below. Then use transformations to graph the function.

 27. $$g(x) = x^{2} + 1$$ 28. $$g(x) = x^{2} − 4$$ 29. $$g(x) = (x − 5)^{2}$$ 30. $$g(x) = (x + 1)^{2}$$ 31. $$g(x) = (x − 5)^{2} + 2$$ 32. $$g(x) = (x + 2)^{2} − 5$$ 33. $$f(t)=(t+1)^2−3$$ 34. $$f ( x ) = - ( x + 2 ) ^ { 2 }$$ 35. $$f ( x ) = - x ^ { 2 } + 6$$  36. $$g ( x ) = - 2 x ^ { 2 }$$ 37. $$g(x)=4(x+1)^2−5$$ 38. $$g(x)=5(x+3)^2−2$$ 39. $$h ( x ) = \tfrac { 1 } { 2 } ( x - 1 ) ^ { 2 }$$ 40. $$h ( x ) = \tfrac { 1 } { 3 } ( x + 2 ) ^ { 2 }$$ 41. $$f ( x ) = ( -\tfrac{1}{2}x - 3 ) ^ 2 + 1$$  42. $$g(x)=(-2x+3)^2 -4$$

$$\bigstar$$ Begin by graphing the square root function $$f(x)=\sqrt{x}$$. State the transformations needed to apply to $$f$$ to graph the function below. Then use transformations to graph the function.

 43. $$g(x) = \sqrt{x} − 5$$ 44. $$g(x) = \sqrt{x − 5}$$ 45. $$g(x) = \sqrt{x − 2} + 1$$ 46. $$g(x) = \sqrt{x + 2} + 3$$ 47. $$a(x)=\sqrt{−x+4}$$ 48. $$m(t)=3-\sqrt{t+2}$$ 49. $$h ( x ) = \sqrt { - x } + 2$$ 50. $$g ( x ) = - \sqrt { x } + 2$$ 51. $$g ( x ) = - \frac { 1 } { 2 } \sqrt { x - 3 }$$ 52. $$h ( x ) = - \sqrt { x - 2 } + 1$$ 53. $$f ( x ) = 4 \sqrt { x - 1 } + 2$$ 54. $$f ( x ) = - 5 \sqrt { x + 2 }$$ 55.  $$k(x) = \sqrt{2x+5} - 1$$ 56.1 $$a(x) = \sqrt{\tfrac{1}{3} x - 4}$$ 56.2 $$b(x) = \sqrt{3-x}+2$$

$$\bigstar$$ Begin by graphing the absolute value function $$f(x)=| x |$$. State the transformations needed to apply to $$f$$ to graph the function below. Then use transformations to graph the function.

 57. $$h(x) = |x + 4|$$ 58. $$h(x) = |x − 4|$$ 59. $$h(x) = |x − 1| − 3$$ 60. $$h(x) = |x + 2| − 5$$ 61. $$g ( x ) = - | x - 1 |$$ 62. $$h(x)=|x−1|+4$$ 63. $$f ( x ) = - 3 | x |$$ 64. $$f ( x ) = - | x | - 3$$ 65. $$h(x)=−2|x−4|+3$$ 66. $$n(x)=\dfrac{1}{3}|x−2|$$ 67. $$h ( x ) = | - 3 x + 4 | - 2$$ 68. $$g(x) = | \tfrac{1}{3}x-2| + 1$$

$$\bigstar$$ Begin by graphing the standard cubic function $$f(x) = x^3$$. State the transformations needed to apply to $$f$$ to graph the function below. Then use transformations to graph the function.

 69. $$h(x) = (x − 2)^{3}$$ 70. $$h(x) = x^{3} + 4$$ 71. $$h(x) = (x − 1)^{3} − 4$$ 72. $$h(x) = (x + 1)^{3} + 3$$ 73. $$g ( x ) = - ( x + 2 ) ^ { 3 }$$ 74. $$k(x)=(x−2)^3−1$$ 75. $$g ( x ) = - x ^ { 3 } + 4$$ 76. $$m(x)=\tfrac{1}{2}x^3$$ 77. $$g ( x ) = - \frac { 1 } { 4 } ( x + 3 ) ^ { 3 } - 1$$ 78. $$q(x)=\big(\tfrac{1}{4}x\big)^3+1$$ 79. $$p(x)=\big(\tfrac{1}{3}x\big)^3−3$$

$$\bigstar$$ Begin by graphing the appropriate parent function : the basic cube root function $$f(x)=\sqrt[3]{x}$$,  constant function $$f(x)=0$$,  or linear function $$f(x)=x$$. Then use transformations of this graph to graph the given function.

 81. $$g( x ) = \sqrt [ 3 ] {x} -1$$ 82. $$g( x ) = \sqrt [ 3 ] { x - 1 }$$ 83. $$g( x ) = \sqrt [ 3 ] { x - 2 } + 6$$ 84. $$g( x ) = \sqrt [ 3 ] { x + 8 } - 4$$ 84.1 $$g( x ) = \sqrt [ 3 ] { -x + 3 } - 2$$ 84.1 $$g( x ) = - \sqrt [ 3 ] { x - 1 } + 2$$ 85. $$g ( x ) = -2 \sqrt [ 3 ] { x + 3 } + 4$$ 86. $$g ( x ) = \sqrt [ 3 ] {-2 x - 5 } - 1$$ 87. $$f(x) = x + 3$$ 88. $$h ( x ) = - 2 x + 1$$ 89. $$g(x) = −4$$

$$\bigstar$$ Begin by graphing the basic reciprocal function $$f(x)=\frac{1}{x}$$. State the transformations needed to apply to $$f$$ to graph the function below. Then use transformations to graph the function.

 91. $$f(x) = \dfrac{1}{x−2}$$ 92. $$f(x) = \dfrac{1}{x+3}$$ 93. $$f(x) = \dfrac{1}{x} + 5$$ 94. $$f(x) = \dfrac{1}{x} − 3$$ 95. $$f(x) = \dfrac{1}{x+1} − 2$$ 96. $$f(x) = \dfrac{1}{x−3} + 3$$ 97. $$f ( x ) = - \dfrac { 1 } { x + 2 }$$ 98. $$f ( x ) = - \dfrac { 1 } { x }$$ 99.$$p( x ) = - \dfrac { 1 } { x + 1 } + 2$$ 100.1 $$a(x) = \dfrac{2}{x-3} -5$$ 100.2 $$b(x) = \dfrac{1}{2x+6} +4$$
Answers to Odd Numbered Exercises for the Squaring Function:
Squaring Function
 27. $$y = x^{2}$$; Shift up $$1$$ unit; domain: $$ℝ$$; range: $$[1, ∞)$$ $$g(x) = x^{2} + 1$$ 29. $$y = x^{2}$$; Shift right $$5$$ units; domain: $$ℝ$$; range: $$[0, ∞)$$ $$g(x) = (x − 5)^{2}$$ 31. $$y = x^{2}$$; Shift right $$5$$ units and up $$2$$ units; domain: $$ℝ$$; range: $$[2, ∞)$$ $$g(x) = (x − 5)^{2} + 2$$ 33.  Shift left $$1$$ unit and down $$3$$ units;  $$f(t)=(t+1)^2−3$$ #35 Reflect over x-axis, up $$6$$ units. 37 $$f(x)=x^2$$ is shifted to the left $$1$$ unit, stretched vertically by a factor of $$4$$, and shifted down $$5$$ units. 39. Shift right $$1$$ unit, and vertically shrink by a factor of $$\frac{1}{2}$$ $$h(x) = \frac { 1 } { 2 } ( x - 1 ) ^ { 2 }$$ #41 Shift right $$3$$, reflect over $$y$$-axis, horizontally stretch by a factor of $$2$$, up $$1$$ units. for # 41, if $$f ( x ) = ( -\tfrac{1}{2}x - 3 ) ^ 2 + 1$$ is rewritten as  $$f ( x ) = ( -\tfrac{1}{2}(x + 6) ) ^ 2 + 1$$ , then the transformations would be horizontal stretch by a factor of 2, reflect in $$y$$-axis (no change), left 6, up 1.
Answers to Odd Numbered Exercises for the Square Root Function

Square Root Function

 43. $$y = \sqrt{x}$$; Shift down $$5$$ units; domain: $$[0, ∞)$$; range: $$[−5, ∞)$$ $$g(x) = \sqrt{x} − 5$$ 45. $$y = \sqrt{x}$$; Shift right $$2$$ units and up $$1$$ unit; domain: $$[2, ∞)$$; range: $$[1, ∞)$$ $$g(x) = \sqrt{x − 2} + 1$$ 47 The graph of $$f(x)=\sqrt{x}$$ is shifted left $$4$$ units and then reflected across the $$y$$-axis.
 49. Reflect over $$y$$-axis, up $$2$$ $$h ( x ) = \sqrt { - x } + 2$$ 51. Right $$3$$, Reflect over $$x$$ axis, Vertically compressed by a factor of $$1/2$$. $$g ( x ) = - \frac { 1 } { 2 } \sqrt { x - 3 }$$ 53. Right 1, Vertically stretched by a factor of $$4$$, up $$2$$ $$f(x) = 4 \sqrt { x - 1 } + 2$$ #55 Horizontal compression by $$1/2$$, shift left $$2.5$$, down $$1$$ unit.
Answers to Odd Numbered Exercises for the Absolute Value Function:

Absolute Value Function

 57. $$y = |x|$$; Shift left $$4$$ units; domain: $$ℝ$$; range: $$[0, ∞)$$ $$h(x) = |x + 4|$$ . 59. $$y = |x|$$; Shift right $$1$$ unit and down $$3$$ units; domain: $$ℝ$$; range: $$[−3, ∞)$$ $$h(x) = |x − 1| − 3$$ . 61. Right $$1$$, Reflect over $$x$$-axis $$g(x) = - | x - 1 |$$ .
 63. Reflect over $$x$$-axis, vertically stretch by a factor of $$3$$ $$f(x) = - 3 | x |$$ 65 The graph of $$f(x)=|x|$$ is shifted horizontally $$4$$ units to the right, stretched vertically by a factor of $$2$$, reflected across the horizontal axis, then shifted up $$3$$ units. 67. $$h(x) = |-3(x-\tfrac{4}{3})| -2$$ $$\longrightarrow$$ Horizontally compress by a factor of $$\tfrac{1}{3}$$, right $$\tfrac{4}{3}$$, down $$2$$
Answers to Odd Numbered Exercises for the Cubing Function

Cubing Function

 69. $$y = x^{3}$$ ; Shift right $$2$$ units; domain: $$ℝ$$; range: $$ℝ$$ $$h(x) = (x − 2)^{3}$$ 71. $$y = x^{3}$$; Shift right $$1$$ unit and down $$4$$ units; domain: $$ℝ$$; range: $$ℝ$$ $$h(x) = (x − 1)^{3} − 4$$ 73. Left $$2$$ units, reflect over x-axis $$g(x) = - ( x + 2 ) ^ { 3 }$$
 75. Reflect over x-axis, up $$4$$ units $$g ( x ) = - x ^ { 3 } + 4$$ 77. Left $$3$$ units, reflect over x-axis, vertically shrink by a factor of $$\frac {1}{4}$$, down $$1$$ unit $$g ( x ) = - \frac {1}{4} (x + 3) ^ {3} - 1$$ 79. Stretch horizontally by a factor of $$3$$ and shift vertically downward by $$3$$ units.
Answers to Odd Numbered Exercises for the Cube Root, Linear and Constant Functions

Cube Root, Linear, Constant Functions

 81. $$y = \sqrt [ 3 ] { x }$$; Shift down $$1$$ 83. $$y = \sqrt [ 3 ] { x }$$; Shift up $$6$$ units and right $$2$$ units; domain: $$ℝ$$; range: $$ℝ$$ 85. $$y = \sqrt [ 3 ] { x }$$; Left $$3$$, reflect over $$x$$-axis, vertically stretch by a factor of $$2$$, up $$4$$.
 87. $$y = x$$; Shift up $$3$$ units; domain: $$\mathbb{R}$$; range: $$\mathbb{R}$$ $$f(x) = x + 3$$ 89. Basic graph $$y = −4$$; domain: $$ℝ$$; range: $$\{−4\}$$ $$g(x) = −4$$
Answers to Odd Numbered Exercises for the Reciprocal Function:

Reciprocal Function

 91. $$y = \frac{1}{x}$$; Shift right $$2$$ units; domain: $$(−∞, 2) ∪ (2, ∞)$$; range: $$(−∞, 0) ∪ (0, ∞)$$ $$f(x) = \frac{1}{x−2}$$ 93. $$y = \frac{1}{x}$$; Shift up $$5$$ units; domain: $$(−∞, 0) ∪ (0, ∞)$$; range: $$(−∞, 1) ∪ (1, ∞)$$ $$f(x) = \frac{1}{x} + 5$$ $$\star$$ 95. $$y = \frac{1}{x}$$; Shift left $$1$$ unit and down $$2$$ units; domain: $$(−∞, −1) ∪ (−1, ∞)$$; range: $$(−∞, −2) ∪ (−2, ∞)$$ $$f(x) = \frac{1}{x+1} − 2$$ 97
 97. Left $$2$$ units, reflect over x-axis $$f ( x ) = - \frac { 1 } { x + 2 }$$ #99  Left $$1$$ unit, reflect over x-axis, up $$2$$ units.

### D: Graph Transformations of a Graph

Exercise $$\PageIndex{D}$$

$$\bigstar$$ Use the graph of $$f(x)$$ shown in the Figure below to sketch a graph of each transformation of $$f(x)$$.

 Given the graph of  $$f(x)$$ on the right, sketch the graph for the following transformations of $$f$$   101. $$h(x)=2^x-3$$ 102. a) $$g(x)=2^x+1$$        b) $$w(x)=2^x−1$$ Given the graph of  $$f(x)$$ on the right, sketch the graph for the following transformations of $$f$$   103. a) $$g(x)=−f(x)$$        b) $$g(x)=f(x−2)$$ 104. a) $$g(x)=f(x)−2$$        b) $$g(x)=f(x+1)$$

106. Given the graph of  $$f(x)$$ below, sketch the graph for the following transformations of $$f$$

 $$a(x) = f(x)+1$$ $$b(x) = f(x+1)$$ $$c(x) = f(x)+2$$ $$d(x) = -f(x)$$ $$e(x) = f(1-x)-2$$ $$f(x) = 2f(x)$$ $$g(x) = -f(x)$$ $$h(x) = \tfrac{1}{2} f(x+2)+3$$ $$i(x) = f(x)-1$$ $$j(x) = f(x-1)$$ $$k(x) = f(x)-2$$ $$l(x) = f(-x)$$ $$m(x) = -f(x-1)+2$$ $$n(x) = f(2x)$$ $$o(x) = f(-x)$$ $$p(x) = f( \tfrac{1}{2} x-2)-3$$
 101 103. a 103. b

### E: Match transformations of functions with graphs

Exercise $$\PageIndex{E}$$

$$\bigstar$$ Match the graph to the function definition.

 107. $$f(x) = \sqrt{x + 4}$$ 108. $$f(x) = |x − 2| − 2$$ 109. $$f(x) = \sqrt{x + 1} -1$$ 110. $$f(x) = |x − 2| + 1$$ 111. $$f(x) = \sqrt{x + 4} + 1$$ 112. $$f(x) = |x + 2| − 2$$

Match the graph to the given function definition.

 113. $$f ( x ) = - 3 | x |$$ 114. $$f ( x ) = - ( x + 3 ) ^ { 2 } - 1$$ 115. $$f ( x ) = - | x + 1 | + 2$$ 116. $$f ( x ) = - x ^ { 2 } + 1$$ 117. $$f ( x ) = - \frac { 1 } { 3 } | x |$$ 118. $$f ( x ) = - ( x - 2 ) ^ { 2 } + 2$$
 Figure (a) Figure (b) $$\star$$ Figure (c) Figure (d) $$\star$$ Figure (e) Figure (f) $$\star$$

part 1 answers 107e, 109d, 111f,    part 2 Answers: 113.b, 115.d, 117.f

### F: Construct equations from graphs of transformed basic functions

Exercise $$\PageIndex{F}$$

$$\bigstar$$ Write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.

 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. (a) 130. (b) 130. (c)
 119. $$f(x)=|x-3|−2$$  121. $$f(x)=\sqrt{x+3}−1$$ 123. $$f(x)=(x-2)^2$$ 125.  $$f(x)=|x+3|−2$$ 127. $$f(x)=−\sqrt{x}$$ 129. $$f(x)=−(x+1)^2+2$$

$$\bigstar$$ Write an equation that represents the function whose graph is given.

 131 132 133 134 135 136 137 138 139
 131. $$f ( x ) = \tfrac{1}{2} \sqrt { x +3 }$$ 133. $$f ( x ) = \sqrt(2x-5)$$ 135. $$f ( x ) = 2 | x-2 | - 3$$ 137. $$f ( x ) = -\tfrac{1}{2} ( x + 2 )^3 +4$$ 139. $$f(x) = \dfrac{1}{x+6} + 4$$

### G: Construct a formula from a description

Exercise $$\PageIndex{G}$$

$$\bigstar$$ Write a formula for the function with the following transformations

141. Write a formula for the function obtained when the graph of $$f(x)=|x|$$ is shifted down $$3$$ units and right $$1$$ unit.

142. Write a formula for the function obtained when the graph of $$f(x)=\dfrac{1}{x}$$ is shifted down $$4$$ units and right $$3$$ units.

143. Write a formula for the function obtained when the graph of $$f(x)=\dfrac{1}{x^2}$$ is shifted up $$2$$ units and left $$4$$ units.

144. Write a formula for the function obtained when the graph of $$f(x)=\sqrt{x}$$ is shifted up $$1$$ unit and left $$2$$ units.

145. The graph of $$f(x)=|x|$$ is reflected over the $$y$$-axis and horizontally compressed by a factor of $$\dfrac{1}{4}$$.

146. The graph of $$f(x)=\sqrt{x}$$ is reflected over the $$x$$-axis and horizontally stretched by a factor of $$2$$.

147. The graph of $$f(x)=\dfrac{1}{x^2}$$ is vertically compressed by a factor of $$\dfrac{1}{3}$$, then shifted left $$2$$ units and down $$3$$ units.

148. The graph of $$f(x)=\dfrac{1}{x}$$ is vertically stretched by a factor of $$8$$, then shifted to the right $$4$$ units and up $$2$$ units.

149. The graph of $$f(x)=x^2$$ is vertically compressed by a factor of $$\dfrac{1}{2}$$, then shifted to the right $$5$$ units and up $$1$$ unit.

150. The graph of $$f(x)=x^2$$ is horizontally stretched by a factor of $$3$$, then shifted left $$4$$ units and down $$3$$ units.

 141. $$g(x)=|x-1|−3$$ 143. $$g(x)=\dfrac{1}{(x+4)^2}+2$$ 145. $$g(x)=|−4x|$$ 147. $$g(x)=\dfrac{1}{3(x+2)^2}−3$$ 149. $$g(x)=\dfrac{1}{2}(x-5)^2+1$$

### H: Construct equations from transformations of tabular values

Exercise $$\PageIndex{H}$$

$$\bigstar$$ Given tabular representations for the functions $$f$$, $$g$$, and $$h$$, write $$g(x)$$ and $$h(x)$$ as transformations of $$f(x)$$.

155. Tabular representations for the functions $$f$$, $$g$$, and $$h$$ are given below. Write $$g(x)$$ and $$h(x)$$ as transformations of $$f(x)$$.

 $$f(x)$$: $$x$$: -2 -1 0 1 2 -2 -1 -3 1 2
 $$x$$: $$g(x)$$: -1 0 1 2 3 -2 -1 -3 1 2
 $$x$$: $$h(x)$$: -2 -1 0 1 2 -1 0 -2 2 3

156. Tabular representations for the functions $$f$$, $$g$$, and $$h$$ are given below. Write $$g(x)$$ and $$h(x)$$ as transformations of $$f(x)$$.

 $$f(x)$$: $$x$$: -2 -1 0 1 2 -1 -3 4 2 1
 $$x$$: $$g(x)$$: -3 -2 -1 0 1 -1 -3 4 2 1
 $$h(x)$$: $$x$$: -2 -1 0 1 2 -2 -4 3 1 0

155. $$g(x)=f(x-1)$$, $$h(x)=f(x)+1$$.

### I: Identify Increasing/decreasing Intervals

Exercise $$\PageIndex{I}$$

$$\bigstar$$ Use transformations to determine the interval(s) on which the function is increasing and decreasing.

 151. $$g(x)=5(x+3)^2−2$$ 152. $$f(x)=4(x+1)^2−5$$ 153. $$k(x)=−3\sqrt{x}−1$$ 154. $$a(x)=\sqrt{−x+4}$$
 151. decreasing on $$(−\infty,−3)$$ and increasing on $$(−3,\infty)$$ 153. decreasing on $$(0, \infty)$$
$$\star$$