Find $$ f(g(x)) $$ and $$ g(f(x)) $$ and determine whether the pair of functions $$ f $$ and $$ g $$ are inverses of each other. $$ f(x)=5 x-7 $$ and $$ g(x)=\frac{x+5}{7} $$ a. $$ f(g(x))=\frac{5 x-24}{7} $$ (Simplify your answer. Use integers or fractions for any numbers in the expression.) b. $$ g(f(x))=\square $$ (Simplify your answer. Use integers or fractions for any numbers in the expression.)

Question

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1. Compute $$ f(g(x)) $$:
   - Substitute $$ g(x)=\frac{x+5}{7} $$ into $$ f $$:
     $$
     f(g(x)) = 5\left(\frac{x+5}{7}\right) - 7
     $$
   - Multiply and simplify:
     $$
     f(g(x)) = \frac{5(x+5)}{7} - 7 = \frac{5x + 25 - 49}{7} = \frac{5x - 24}{7}
     $$
     
2. Compute $$ g(f(x)) $$:
   - Substitute $$ f(x)=5x-7 $$ into $$ g $$:
     $$
     g(f(x)) = \frac{(5x-7)+5}{7}
     $$
   - Simplify the numerator:
     $$
     g(f(x)) = \frac{5x - 7 + 5}{7} = \frac{5x - 2}{7}
     $$
     
3. Determine whether $$ f $$ and $$ g $$ are inverses:
   - For $$ f $$ and $$ g $$ to be inverses, we must have:
     $$
     f(g(x)) = x \quad \text{and} \quad g(f(x)) = x
     $$
   - We found:
     $$
     f(g(x)) = \frac{5x-24}{7} \quad \text{and} \quad g(f(x)) = \frac{5x-2}{7}
     $$
   - These expressions equal $$ x $$ only for specific values of $$ x $$ (for example, $$ f(g(x)) = x $$ if $$ \frac{5x-24}{7}=x $$ and $$ g(f(x)) = x $$ if $$ \frac{5x-2}{7}=x $$), not for all $$ x $$.
   - Therefore, $$ f $$ and $$ g $$ are not inverses of each other.
   
Final answers:
- $$ f(g(x))=\frac{5x-24}{7} $$
- $$ g(f(x))=\frac{5x-2}{7} $$
- The functions $$ f $$ and $$ g $$ are not inverses of each other.
$$ f(g(x)) = \frac{5x - 24}{7} $$ $$ g(f(x)) = \frac{5x - 2}{7} $$ The functions $$ f $$ and $$ g $$ are not inverses of each other.

Find and and determine whether the pair of functions and are inverses of each other.
and
a.
(Simplify your answer. Use integers or fractions for any numbers in the expression.)
b.
(Simplify your answer. Use integers or fractions for any numbers in the expression.)

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b. Finding

Checking if and are inverses

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Find and and determine whether the pair of functions and are inverses of each other. and a. (Simplify your answer. Use integers or fractions for any numbers in the expression.) b. (Simplify your answer. Use integers or fractions for any numbers in the expression.)