5. Domain Restriction for Inverse Functions

Multiple Choice

Was the inverse function found correctly?

• no,

• yes

The functions f(x) and g(x) are graphed together on the same plane above. Since f and g are inverses, they should be symmetric across the line y=x.
Because this is not a mirror image across y=x like it should be, we must restrict the domain of the parabola.

If we remove the left side of the parabola, we will have the same graph on each side of the line y=x.

​f(x)

​g(x)

​f(x)

​g(x)

When we remove half of the parabola, we successfully restrict the domain of f(x) in order for f and g to be inverses.

​

​f(x)

​g(x)

Multiple Choice

What is the restricted domain of f(x)?

• $$\left(-\infty,0\right)$$

• $$\left[0,\ \infty\right)$$

• all real numbers

• $$\left[2,\ \infty\right)$$

Multiple Choice

What is the range of g(x)?

• $$\left(-\infty,0\right)$$

• $$\left[0,\ \infty\right)$$

• all real numbers

• $$\left[2,\ \infty\right)$$

Multiple Choice

What is the domain of g(x)?

• $$\left(-\infty,0\right)$$

• $$\left[0,\ \infty\right)$$

• all real numbers

• $$\left[2,\ \infty\right)$$

Multiple Choice

What is the range of f(x)?

• $$\left(-\infty,0\right)$$

• $$\left[0,\ \infty\right)$$

• all real numbers

• $$\left[2,\ \infty\right)$$

Fill in the Blank

The domain of f(x) is equal to the ____ of its inverse.

Fill in the Blank

The range of f(x) is equal to the ____ of its inverse.

Multiple Choice

Is the graph a function? Hint: use the vertical line test.

• yes

• no

Multiple Choice

Is the graph a function? Hint: use the vertical line test.

• yes

• no

​The horizontal line test tells us if the graph's inverse is a function without having to actually find the inverse.

It works the same way as the vertical line test.

If the graph touches a horizontal line more than once, its inverse is not a function.

Multiple Choice

Is the graph's inverse a function? Hint: use the horizontal line test.

• yes

• no

Multiple Choice

Is the graph's inverse a function? Hint: use the horizontal line test.

• yes

• no

Generating the inverse of Quadratics

​

Math Response

Find the inverse of  $$f\left(x\right)=x^2-5$$  if the domain is restricted to  $$x\ge0$$  

$$f^{-1}\left(x\right)=$$ ?

Example answers: $$-\sqrt[]{x-2}+3$$ , $$\sqrt[]{x+2}-5$$

Type answer here

Correct Answer: \sqrt{x+5}

Deg^°

Rad

Math Response

Find the inverse of  $$f\left(x\right)=x^2-8$$  if the domain is restricted to  $$x\le0$$  

$$f^{-1}\left(x\right)=$$ ?

Example answers: $$-\sqrt[]{x-2}+3$$ , $$\sqrt[]{x+2}-5$$

Type answer here

Correct Answer: -\sqrt{x+8}

Deg^°

Rad

Math Response

Find the inverse of  $$f\left(x\right)=\left(x-2\right)^2+3$$  if the domain is restricted to  $$x\ge2$$  

$$f^{-1}\left(x\right)=$$ ?

Example answers: $$-\sqrt[]{x-2}+3$$ , $$\sqrt[]{x+2}-5$$

Type answer here

Correct Answer: \sqrt{x-3}+2

Deg^°

Rad

Graphing

The graph displays a linear function that includes the points $$\left(-5,\ 1\right)$$ and $$\left(7,\ -1\right)$$ .

Drag the points below to create the inverse of the function.

Graphing

The exponential function $$f\left(x\right)$$ contains the ordered pairs $$\left(-1,\ -1\right),\ \left(4,\ 2\right),\ \left(6,\ 5\right),\ and\ \left(8,\ 10\right)$$ .

Plot four ordered pairs to represent $$f^{-1}\left(x\right)$$ .