Question

Show that the function g(x)=x-2/5 is the inverse of f(x)=5x+2Step 1: the function notation f(x) can be written as a variable in an equation. Is that variable x or y?Write f(x)=5x+2 as an equation with the variable you chose above.Step 2: switch the variables in the equation from Step 1. Then solve for y. Show your work.Step 3: Find the inverse of g(x)= x-2/5. What does this tell you about the relationship between f(x)=5x+2 and g(x)? Show your work.

Answer 1

Given that :

$f(x)\text{ = 5x + 2}$

We can prove that :

$g(x)\text{ = }\frac{x\text{ -2}}{5}$

is it's inverse doing the following:

Step 1. Set y = f(x):

$y\text{ = 5x + 2}$

Step 2. Switch the variables:

$x\text{ = 5y + 2}$

Then we solve for y:

$\begin{gathered} 5y\text{ = x - 2} \\ \text{Divide both sides by 5} \\ y\text{ = }\frac{x\text{ -2}}{5} \end{gathered}$

Step 3. The inverse of :

$g(x)\text{ = }\frac{x-2}{5}$

can be found in a similar way.

$\begin{gathered} y\text{ = }\frac{x-2}{5} \\ x\text{ = }\frac{y-2}{5} \\ \text{Cross}-\text{Multiply} \\ 5x\text{ = y -2} \\ y\text{ = 5x + 2} \end{gathered}$

This tells us that f(x) and g(x) are one to one functions are f(x) is the mirror image of g(x)