### About Determine if Two Functions are Inverses:

If two functions f(x) and g(x) are inverses, then it must be true that f(g(x)) = x and g(f(x)) = x. If one of these conditions fails, we can report that they are not inverses. We must check both in order to say that our two functions f and g are inverses.

Test Objectives
• Demonstrate an understanding of function composition
• Demonstrate the ability to determine if two functions are inverses
Determine if Two Functions are Inverses Practice Test:

#1:

Instructions: determine if the given functions f(x) and g(x) are inverses.

$$a)\hspace{.2em}$$ $$f(x)=\frac{3}{x - 1}- 1$$ $$g(x)=\frac{3}{x + 1}+ 1$$

$$b)\hspace{.2em}$$ $$f(x)=\frac{4}{x}- 3$$ $$g(x)=\frac{4}{x + 3}$$

#2:

Instructions: determine if the given functions f(x) and g(x) are inverses.

$$a)\hspace{.2em}$$ $$f(x)=\frac{1}{x - 1}$$ $$g(x)=\frac{4}{x - 2}- 1$$

$$b)\hspace{.2em}$$ $$f(x)=(x + 1)^3 + 1$$ $$g(x)=\sqrt{x - 1}- 1$$

#3:

Instructions: determine if the given functions f(x) and g(x) are inverses.

$$a)\hspace{.2em}$$ $$f(x)=\sqrt{x - 2}- 2$$ $$g(x)=-x^3 + 4$$

$$b)\hspace{.2em}$$ $$f(x)=1 + \frac{1}{2}x$$ $$g(x)=2x - 2$$

#4:

Instructions: determine if the given functions f(x) and g(x) are inverses.

$$a)\hspace{.2em}$$ $$f(x)=\sqrt{x - 1}$$ $$g(x)=x^5 + 1$$

$$b)\hspace{.2em}$$ $$f(x)=-\frac{\sqrt{4x}}{2}$$ $$g(x)=\sqrt{x - 2}+ 2$$

#5:

Instructions: determine if the given functions f(x) and g(x) are inverses.

$$a)\hspace{.2em}$$ $$f(x)=x + 5$$ $$g(x)=x - 5$$

$$b)\hspace{.2em}$$ $$f(x)=\frac{x - 3}{x + 7}$$ $$g(x)=\frac{x - 7}{x + 3}$$

Written Solutions:

#1:

Solutions:

a) Inverses

b) Inverses

#2:

Solutions:

a) Not Inverses

b) Inverses

#3:

Solutions:

a) Not Inverses

b) Inverses

#4:

Solutions:

a) Inverses

b) Not Inverses

#5:

Solutions:

a) Inverses

b) Not Inverses