3.2.1: Exercises 3.2

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Terms and Concepts

Exercise $$\PageIndex{1}$$

In which situations is substitution a more appropriate solution method than equating the functions?

When one or both functions are defined implicitly.

Exercise $$\PageIndex{2}$$

Is $$y=x^3+5x-7$$ an implicitly or explicitly defined function? Explain.

Explicitly; $$y$$ is isolated.

Exercise $$\PageIndex{3}$$

Is $$xy+y^2 -y = 2x+6$$ an implicitly or explicitly defined function? Explain.

Implicitly; $$y$$ is not isolated.

Exercise $$\PageIndex{4}$$

Describe the pros and cons of using graphing to find the point(s) of intersection.

Answers will vary, but graphing helps you determine how many intersections points exist, but does not always clearly show the exact values.

Problems

In exercises $$\PageIndex{5}$$ - $$\PageIndex{8}$$, determine the maximum possible number of intersections for the described functions.

Exercise $$\PageIndex{5}$$

Two linear functions with different slopes

1

Exercise $$\PageIndex{6}$$

A linear function and a quadratic function

2

Exercise $$\PageIndex{7}$$

2

Exercise $$\PageIndex{8}$$

A cubic function and a constant function

3

In exercises $$\PageIndex{9}$$ - $$\PageIndex{12}$$, determine the minimum possible number of intersections for the described functions.

Exercise $$\PageIndex{9}$$

Two linear functions with different slopes

1

Exercise $$\PageIndex{10}$$

A linear function and a quadratic function

0

Exercise $$\PageIndex{11}$$

0

Exercise $$\PageIndex{12}$$

A cubic function and a constant function

1

In exercises $$\PageIndex{13}$$ - $$\PageIndex{18}$$, find all points of intersection between the given functions.

Exercise $$\PageIndex{13}$$

$$y=x^2-1$$ and $$y=x-1$$

$$(0,-1)$$ and $$(1,0)$$

Exercise $$\PageIndex{14}$$

$$x^2+y^2=1$$ and $$4y=3x$$

$$(\frac{4}{5},\frac{3}{5})$$ and $$(-\frac{4}{5}, -\frac{3}{5})$$

Exercise $$\PageIndex{15}$$

$$y-1 = \sqrt{3x}$$ and $$y=x+1$$

$$(0,1)$$ and $$(3,4)$$

Exercise $$\PageIndex{16}$$

$$y=x^2-3x+2$$ and the x-axis

$$(1,0)$$ and $$(2,0)$$

Exercise $$\PageIndex{17}$$

$$y=x^2-3x+2$$ and $$y=5$$

$$(\frac{3+\sqrt{21}}{2},5)$$ and $$(\frac{3-\sqrt{21}}{2},5)$$

Exercise $$\PageIndex{18}$$

$$y+2x=5$$ and $$y+3=x^3-7x^2+12x$$

$$(1,3)$$, $$(2,1)$$, and $$(4,-3)$$

In exercises $$\PageIndex{19}$$ - $$\PageIndex{22}$$, sketch the region bounded by the given functions and determine all intersection points.

Exercise $$\PageIndex{19}$$

$$y=x^2$$ and $$y=x$$

Points of intersection are $$(0,0)$$ and $$(1,1)$$

Exercise $$\PageIndex{20}$$

$$y=x^2$$ and $$y=x+2$$

Points of intersection are $$(-1,1)$$ and $$(2,4)$$

Exercise $$\PageIndex{21}$$

$$y=x^2$$ and $$y=\sqrt{x}$$

Points of intersection are $$(0,0)$$ and $$(1,1)$$

Exercise $$\PageIndex{22}$$

$$3y+2x=6$$, the x-axis, and the y-axis (hint: sketch before looking for the intersection points)

Points of intersection are $$(0,0)$$, $$(0,2)$$, and $$(3,0)$$