Question

. Find the inverse of the function below on the given interval and write it in the form yequalsf Superscript negative 1 Baseline (x ). b. Verify the relationships f (f Superscript negative 1 Baseline (x ))equalsx and f Superscript negative 1 Baseline (f (x ))equalsx.

Answer 1

Answer:

The inverse of the function is $f^{-1}(x)=\frac{x-5}{3}$.

Step-by-step explanation:

The function provided is:

$f (x)=3x+5$

Let $f(x)=y$.

Then the value of x is:

$y=3x+5\\\\3x=y-5\\\\x=\frac{y-5}{3}$

For the inverse of the function, $x\rightarrow y$.

⇒ $f^{-1}(x)=\frac{x-5}{3}$

Compute the value of $f[f^{-1}(x)]$ as follows:

$f[f^{-1}(x)]=f[\frac{x-5}{3}]$

               $=3[\frac{x-5}{3}]+5\\\\=x-5+5\\\\=x$

Hence proved that $f[f^{-1}(x)]=x$.

Compute the value of $f^{-1}[f(x)]$ as follows:

$f^{-1}[f(x)]=f^{-1}[3x+5]$

               $=\frac{(3x+5)-5}{3}\\\\=\frac{3x+5-5}{3}\\\\=x$

Hence proved that $f^{-1}[f(x)]=x$.