6.2 Inverse Functions and relations

6.2 Inverse Functions and relations

Fill in the Blank

 $f\left(x\right)=3x+2$  and  $g\left(x\right)=2x^2-1$  .  Find (g - f)(x)

Fill in the Blank

 $f\left(x\right)=3x+2$  and  $g\left(x\right)=2x^2-1$  .  Find  $\left(f\cdot g\right)\left(x\right)$  

Fill in the Blank

 $f\left(x\right)=3x+2$  and  $g\left(x\right)=2x^2-1$  .  Find  $\left(g\circ f\right)\left(x\right)$  

Multiple Choice

 $f\left(x\right)=x-3$  and  $g\left(x\right)=x^2$ .  Evaluate  $\left(f\circ g\right)\left(1\right)$   

• 1

• -2

• 2

• 4

Find the inverse of

 $f\left(x\right)=-\frac{1}{2}x+1$  

• Step 1:  replace f(x) in the original equation with y

•  $y=-\frac{1}{2}x+1$  

• Step 2: interchange x and y

•  $x=-\frac{1}{2}y+1$  

• Step 3: Solve for y

•  $x-1=-\frac{1}{2}y$  

•  $-2\left(x-1\right)=y$  

• -2x+2=y

• Step 4: Replace y with  $f^{-1}\left(x\right)$  

•  $f^{-1}\left(x\right)=-2x+2$  

Original

 $f\left(x\right)=\frac{1}{3}x+6$  

Inverse:  $f^{-1}\left(x\right)=3x-18$  

Find the inverse of the function graph the function and its inverse.  

 $f\left(x\right)=x^2+1$  

•  $y=x^2+1$  

•  $x=y^2+1$  

•  $x-1=y^2$  

•  $y=\pm\sqrt{x-1}$  

•  $f^{-1}\left(x\right)=\pm\sqrt{x-1}$  

 $f\left(x\right)=x^2+1$  

• that is a parabola you know how to graph

• But how to graph a square root function?

• Remember that its the inverse.  That means the x and y values were interchanged!

Determine whether f(x) and g(x) are inverses of each other.

 $f\left(x\right)=\frac{3}{4}x-6$  and  $g\left(x\right)=\frac{4}{3}x+8$  

• You must show  $\left(f\circ g\right)\left(x\right)=x$   AND  $\left(g\circ f\right)\left(x\right)=x$  

 $\left(f\circ g\right)\left(x\right)=x$  

•  $f\left(\frac{4}{3}x+8\right)=x$  

•  $\frac{3}{4}\left(\frac{4}{3}x+8\right)-6=x$  

•  $x+6-6=x$  

• x=x

• YOU ARE ONLY HALF DONE.  NOW  $\left(g\circ f\right)\left(x\right)=x$  

 $\left(g\circ f\right)\left(x\right)=x$  

•  $g\left(\frac{3}{4}x-6\right)=x$  

•  $\frac{4}{3}\left(\frac{3}{4}x-6\right)+8=x$  

•  $x-8+8=x$  

•  $x=x$  

• Yes, f(x) and g(x) are inverse of each other

Multiple Choice

Determine whether $f\left(x\right)=3x-1$  and  $g\left(x\right)=\frac{x-1}{3}$  are inverses of each other.  Use composition of functions.

• Yes

• No