Find $$ f(g(x)) $$ and $$ g(f(x)) $$ and determine whether the pair of functior $$ f(x)=7 x-8 $$ and $$ g(x)=\frac{x+7}{8} $$ a. $$ f(g(x))=\square $$ (Simplify your answer. Use integers or fractions for any numbers b. $$ g(f(x))=\square $$ (Simplify your answer. Use integers or fractions for any numbers c. $$ \quad f $$ and $$ g $$ are inverses of each other. c. $$ \quad f $$ and $$ g $$ are not inverses of each other.

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UpStudy AI · Accepted Answer

**Step 1. Compute $$ f(g(x)) $$:**

Given
$$
f(x)=7x-8 \quad \text{and} \quad g(x)=\frac{x+7}{8},
$$
substitute $$ g(x) $$ into $$ f(x) $$:
$$
f(g(x))=7\left(\frac{x+7}{8}\right)-8.
$$
Multiply:
$$
= \frac{7(x+7)}{8} - 8.
$$
Write $$8$$ as $$\frac{64}{8}$$ to combine:
$$
= \frac{7x+49-64}{8} = \frac{7x-15}{8}.
$$

**Step 2. Compute $$ g(f(x)) $$:**

Substitute $$ f(x) $$ into $$ g(x) $$:
$$
g(f(x))=\frac{f(x)+7}{8} = \frac{(7x-8)+7}{8}.
$$
Simplify the numerator:
$$
= \frac{7x-1}{8}.
$$

**Step 3. Determine if $$ f $$ and $$ g $$ are inverses:**

For two functions to be inverses, both compositions must equal the identity function:
$$
f(g(x))=x \quad \text{and} \quad g(f(x))=x.
$$

We found:
$$
f(g(x))=\frac{7x-15}{8} \quad \text{and} \quad g(f(x))=\frac{7x-1}{8}.
$$

Since neither $$ \frac{7x-15}{8} $$ nor $$ \frac{7x-1}{8} $$ equals $$ x $$ for all $$ x $$, the functions $$ f $$ and $$ g $$ are not inverses of each other.

**Final Answers:**

a. $$ f(g(x))=\frac{7x-15}{8} $$  
b. $$ g(f(x))=\frac{7x-1}{8} $$  
c. $$ f $$ and $$ g $$ are not inverses of each other.
a. $$ f(g(x)) = \frac{7x - 15}{8} $$ b. $$ g(f(x)) = \frac{7x - 1}{8} $$ c. $$ f $$ and $$ g $$ are not inverses of each other.

Find and and determine whether the pair of functior
and
a.
(Simplify your answer. Use integers or fractions for any numbers
b.
(Simplify your answer. Use integers or fractions for any numbers
c. and are inverses of each other.
c. and are not inverses of each other.

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Find and and determine whether the pair of functior and a. (Simplify your answer. Use integers or fractions for any numbers b. (Simplify your answer. Use integers or fractions for any numbers c. and are inverses of each other. c. and are not inverses of each other.