(b) $$ f(x)=\frac{1}{4 x}, x
eq 0 $$ $$ g(x)=-\frac{1}{4 x}, x
eq 0 $$ $$ f(g(x))=\square $$ $$ g(f(x))=\square $$ $$ f $$ and $$ g $$ are inverses of each other $$ f $$ and $$ g $$ are not inverses of each other

Question

(b) $$ f(x)=\frac{1}{4 x}, x 
eq 0 $$ $$ g(x)=-\frac{1}{4 x}, x 
eq 0 $$ $$ f(g(x))=\square $$ $$ g(f(x))=\square $$ $$ f $$ and $$ g $$ are inverses of each other $$ f $$ and $$ g $$ are not inverses of each other

UpStudy AI · Accepted Answer

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

Given  
$$
f(x)=\frac{1}{4x} \quad \text{and} \quad g(x)=-\frac{1}{4x},
$$
we substitute $$ g(x) $$ into $$ f $$:
$$
f(g(x)) = f\Bigl(-\frac{1}{4x}\Bigr) = \frac{1}{4\Bigl(-\frac{1}{4x}\Bigr)}.
$$
Simplify the denominator:
$$
4\Bigl(-\frac{1}{4x}\Bigr) = -\frac{1}{x}.
$$
Thus,
$$
f(g(x)) = \frac{1}{-\frac{1}{x}} = -x.
$$

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

Substitute $$ f(x) $$ into $$ g $$:
$$
g(f(x)) = g\Bigl(\frac{1}{4x}\Bigr) = -\frac{1}{4\Bigl(\frac{1}{4x}\Bigr)}.
$$
Simplify the expression in the denominator:
$$
4\Bigl(\frac{1}{4x}\Bigr) = \frac{1}{x}.
$$
Thus,
$$
g(f(x)) = -\frac{1}{\frac{1}{x}} = -x.
$$

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

For two functions to be inverses, the compositions must satisfy:
$$
f(g(x)) = x \quad \text{and} \quad g(f(x)) = x.
$$
Since we obtained
$$
f(g(x)) = -x \quad \text{and} \quad g(f(x)) = -x,
$$
which do not equal $$ x $$ for all $$ x $$, the functions $$ f $$ and $$ g $$ are not inverses of each other.

**Final Answers:**

$$
f(g(x)) = -x
$$
$$
g(f(x)) = -x
$$
$$
\text{$$ f $$ and $$ g $$ are not inverses of each other.}
$$
$$ f(g(x)) = -x $$, $$ g(f(x)) = -x $$, and $$ f $$ and $$ g $$ are not inverses of each other.

(b)

and are inverses of each other
and are not inverses of each other

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(b) and are inverses of each other and are not inverses of each other