1.1E: Exercises
Exercises
Verbal
- What is the difference between a relation and a function?
- What is the difference between the input and the output of a function?
- Why does the Vertical Line Test tell us whether the graph of a relation represents a function?
- How can you determine if a relation is a one-to-one function?
- Why does the Horizontal Line Test tell us whether the graph of a function is one-to-one?
Algebraic
For the following exercises, determine whether the relation represents a function.
- \(\{(a,b),(c,d),(a,c)\}\)
- \(\{(a,b),(b,c),(c,c)\}\)
For the following exercises, determine whether the relation represents \( y \) as a function of \( x \).
- \( 5x + 2y = 10 \)
- \( y = x^2 \)
- \( x = y^2 \)
- \( 3x^2 + y = 14 \)
- \( 2x + y^2 = 6 \)
- \( y = -2x^2 + 40x \)
- \( y = \frac{1}{x} \)
- \( x = \frac{3y + 5}{7y - 1} \)
- \( x = \sqrt{1 - y^2} \)
- \( y = \frac{3x + 5}{7x - 1} \)
- \( x^2 + y^2 = 9 \)
- \( 2xy = 1 \)
- \( x = y^3 \)
- \( y = x^3 \)
- \( y = \sqrt{1 - x^2} \)
- \( x = \pm \sqrt{1 - y} \)
- \( y = \pm \sqrt{1 -x} \)
- \( y^2 = x^2 \)
- \( y^3 = x^2 \)
For the following exercises, evaluate \( f(-3) \), \( f(2) \), \( f(-a) \), \( -f(a) \), and \( f(a + h) \).
- \( f(x) = 2x - 5 \)
- \( f(x) = -5x^2 + 2x -1 \)
- \( f(x) = \sqrt{2 - x} + 5 \)
- \( f(x) = \frac{6x - 1}{5x + 2} \)
- \( f(x) = |x - 1| - |x + 1| \)
- Given the function \( g(x) = 5 - x^2 \), evaluate\[ \dfrac{g(x + h) - g(x)}{h}, \text{ where } h \neq 0. \nonumber \]
- Given the function \( g(x) = x^2 + 2x \), evaluate\[ \dfrac{g(x) - g(a)}{x - a}, \text{ where } x \neq a. \nonumber \]
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Given the function \( k(t)=2t−1 \):
- Evaluate \( k(2) \).
- Solve \( k(t)=7 \).
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Given the function \( f(x)=8−3x \):
- Evaluate \( f(−2) \).
- Solve \( f(x)=−1 \).
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Given the function \( p(c)=c^2+c \):
- Evaluate \( p(−3) \).
- Solve \( p(c)=2 \).
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Given the function \( f(x)=x^2−3x \):
- Evaluate \( f(5) \).
- Solve \( f(x)=4 \).
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Given the function \( f(x)=\sqrt{x+2} \):
- Evaluate \( f(7) \).
- Solve \( f(x)=4 \).
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Consider the relationship \( 3r+2t=18 \).
- Write the relationship as a function \( r=f(t) \).
- Evaluate \( f(−3) \).
- Solve \( f(t)=2 \).
Graphical
For the following exercises, use the Vertical Line Test to determine which graphs show relations that are functions.
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Given the following graph,
- Evaluate \( f(−1) \).
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Solve for \( f(x)=3 \).
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Given the following graph,
- Evaluate \( f(0) \).
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Solve for \( f(x)=−3 \).
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Given the following graph,
- Evaluate \( f(4) \).
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Solve for \( f(x)=1 \).
For the following exercises, determine if the given graph is a one-to-one function.
Numeric
For the following exercises, determine whether the relation represents a function.
- \( \{ ( −1 , −1 ) , ( −2 , −2 ) , ( −3 , −3 ) \} \)
- \( \{ ( 3 , 4 ) , ( 4 , 5 ) , ( 5 , 6 ) \} \)
- \( \{ ( 2 , 5 ) , ( 7 , 11 ) , ( 15 , 8 ) , ( 7 , 9 ) \} \)
For the following exercises, determine if the relation represented in table form represents \( y \) as a function of \( x \).
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\( x \) 5 10 15 \( y \) 3 8 14 -
\( x \) 5 10 15 \( y \) 3 8 8 -
\( x \) 5 10 10 \( y \) 3 8 14
For the following exercises, use the function \( f \) represented in the following table.
| \( x \) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| \( f(x) \) | 74 | 28 | 1 | 53 | 56 | 3 | 36 | 45 | 14 | 47 |
- Evaluate \( f ( 3 ) \).
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Solve \( f ( x ) = 1 \).
For the following exercises, evaluate the function \( f \) at the values \( -2 \), \( -1 \), \( 0 \), \( 1 \), and \( 2 \).
- \( f ( x ) = 4 − 2 x \)
- \( f ( x ) = 8 − 3 x \)
- \( f ( x ) = 8 x^2 − 7 x + 3 \)
- \( f ( x ) = 3 + \sqrt{x + 3} \)
- \( f ( x ) = \frac{x − 2}{x + 3} \)
- \( f ( x ) = 3^x \)
For the following exercises, evaluate the expressions, given functions \( f \), \( g \), and \( h \):
- \( f ( x ) = 3 x − 2 \)
- \( g ( x ) = 5 − x^2 \)
- \( h ( x ) = − 2 x^2 + 3 x − 1 \)
- \( 3 f ( 1 ) − 4 g ( − 2 ) \)
- \( f \left( \frac{7}{3} \right) − h ( − 2 ) \)
Technology
For the following exercises, graph \( y = x^2 \) on the given domain. Determine the corresponding range. Show each graph.
- \( [ − 0.1 , 0.1 ] \)
- \( [ − 10 , 10 ] \)
- \( [ − 100 , 100 ] \)
For the following exercises, graph \( y = x^3 \) on the given domain. Determine the corresponding range. Show each graph.
- \( [ − 0.1 , 0 .1 ] \)
- \( [ − 10 , 10 ] \)
- \( [ − 100 , 100 ] \)
For the following exercises, graph \( y = \sqrt{x} \) on the given domain. Determine the corresponding range. Show each graph.
- \( [ 0 , 0 .01 ] \)
- \( [ 0 , 100 ] \)
- \( [ 0 , 10000 ] \)
For the following exercises, graph \( y = \sqrt[3]{ x} \)
- \( [ −0.001 , 0.001 ] \)
- \( [ −1000 , 1000 ] \)
- \( [ −1000000 , 1000000 ] \)
Real-World Applications
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The amount of garbage, \( G \), produced by a city with population \( p \) is given by \( G = f ( p ) \). \( G \) is measured in tons per week, and \( p \) is measured in thousands of people.
- The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function \( f \).
- Explain the meaning of the statement \( f ( 5 ) = 2 \).
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The number of cubic yards of dirt, \( D \), needed to cover a garden with area \( a \) square feet is given by \( D = g ( a ) \).
- A garden with area 5000 ft 2 requires 50 yd 3 of dirt. Express this information in terms of the function \( g \).
- Explain the meaning of the statement \( g ( 100 ) = 1 \).
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Let \( f ( t ) \) be the number of ducks in a lake \( t \) years after 1990. Explain the meaning of each statement:
- \( f ( 5 ) = 30 \)
- \( f ( 10 ) = 40 \)
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Let \( h ( t ) \) be the height above ground, in feet, of a rocket \( t \) seconds after launching. Explain the meaning of each statement:
- \( h ( 1 ) = 200 \)
- \( h ( 2 ) = 350 \)
- Show that the function \( f ( x ) = 3 ( x − 5 )^2 + 7 \) is not one-to-one.