Question

Verify that the functions are inverse functions.$f\left(x\right)=4x-3\mathrm{and}g\left(x\right)=\frac{x+3}{4}.$

Short answer

Since both g(f(x)) = x and f(g(x)) = x are true ,the functions f(x)=4x-3 and g(x)= $\frac{x+3}{4}$are inverse functions. That is, they are inverses of each other.

Step-by-step solution

Step 1. Given information

The given equation are $f\left(x\right)=4x-3\mathrm{and}g\left(x\right)=\frac{x+3}{4}.$

Step 2.  Checking for the equation first

The functions are inverses of each other if g(f(x)= x and f(g(x) = x.

$$\begin{gathered} \mathrm{Subsitute}4\mathrm{x}-3\mathrm{for}\mathrm{f}\left(\mathrm{x}\right)\mathrm{g}(4\mathrm{x}-3)=? \\ \mathrm{Find}\mathrm{g}(4\mathrm{x}-3)\mathrm{where}g\left(x\right)=\frac{x+3}{4}\frac{4x-3+3}{4} \\ =\frac{4x}{4} \\ =x=x \end{gathered}$$

Step 3. Checking for the equation second

$$\begin{gathered} \mathrm{Substitute}\frac{\mathrm{x}+3}{4}\mathrm{for}\mathrm{g}\left(\mathrm{x}\right).\mathrm{f}\left(\frac{\mathrm{x}+3}{4}\right)=\mathrm{x} \\ \mathrm{Find}\mathrm{f}\left(\frac{\mathrm{x}+3}{4}\right)\mathrm{for}f\left(x\right)=4x-34(\frac{x+3}{4})-3 \\ =x+3-3 \\ =x \end{gathered}$$

Since both g(f(x)) = x and f(g(x)) = x are true ,the functions f(x)=4x-3 and g(x)= $\frac{x+3}{4}$are inverse functions. That is, they are inverses of each other.