Question

For f(x) = 1/x-5 and g(x)= x² +2, find:Part 1:A. Find the expression for g(x). B. Substitute the value of g(x) into the function f(x) in place of x to find thevalue of f(g(x)). Part II: (gof)(6)A. Find f(6). B. Substitute the value you found in Part I into g(x) to find g(f(6)).

Answer 1

The Solution:

Given:

$\begin{gathered} f(x)=\frac{1}{x-5} \\ \\ g(x)=x^2+2 \end{gathered}$

Required:

Part II:

Find the values of:

$gof(6)$

Step 1:

Find g(f(x) by substituting f(x) in the place x in g(x).

$g(f(x))=(\frac{1}{x-5})^2+2$$g(f(x))=\frac{1}{(x-5)^2}+2$

Step 2:

Find the value of g(f(6)).

Substitute x = 6 in g(f(x)).

$g(f(6))=\frac{1}{(6-5)^2}+2=\frac{1}{1^2}+2=1+2=3$

Alternatively:

Substitute x = 6 in f(x).

$f(6)=\frac{1}{6-5}=\frac{1}{1}=1$

Substitute f(6) = 1 in g(x) to get g(f(6)).

$g(f(6))=g(1)=1^2+2=1+2=3$

Answer:

g(f(6)) = 3