Are the following functions inverses? $$ \begin{array}{l}f(x)=4 x-2 \ g(x)=\frac{x}{4}+2 \ ext { No, they are not inverses } \ ext { Yes, they are inverses }\end{array} $$

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Are the following functions inverses? $$ \begin{array}{l}f(x)=4 x-2 \ g(x)=\frac{x}{4}+2 \ 	ext { No, they are not inverses } \ 	ext { Yes, they are inverses }\end{array} $$

UpStudy AI · Accepted Answer

To determine if the functions $$ f(x) = 4x - 2 $$ and $$ g(x) = \frac{x}{4} + 2 $$ are inverses of each other, we need to check if the following conditions hold:

1. $$ f(g(x)) = x $$
2. $$ g(f(x)) = x $$

Let's calculate $$ f(g(x)) $$ and $$ g(f(x)) $$ step by step.

### Step 1: Calculate $$ f(g(x)) $$

Substituting $$ g(x) $$ into $$ f(x) $$:

$$
f(g(x)) = f\left(\frac{x}{4} + 2\right)
$$

Now, substitute $$ \frac{x}{4} + 2 $$ into $$ f(x) $$:

$$
f\left(\frac{x}{4} + 2\right) = 4\left(\frac{x}{4} + 2\right) - 2
$$

Now simplify:

$$
= 4 \cdot \frac{x}{4} + 4 \cdot 2 - 2
$$
$$
= x + 8 - 2
$$
$$
= x + 6
$$

### Step 2: Calculate $$ g(f(x)) $$

Now, substitute $$ f(x) $$ into $$ g(x) $$:

$$
g(f(x)) = g(4x - 2)
$$

Now, substitute $$ 4x - 2 $$ into $$ g(x) $$:

$$
g(4x - 2) = \frac{4x - 2}{4} + 2
$$

Now simplify:

$$
= x - \frac{1}{2} + 2
$$
$$
= x + \frac{3}{2}
$$

### Conclusion

We found:

- $$ f(g(x)) = x + 6 $$
- $$ g(f(x)) = x + \frac{3}{2} $$

Since neither $$ f(g(x)) $$ nor $$ g(f(x)) $$ equals $$ x $$, the functions $$ f(x) $$ and $$ g(x) $$ are **not inverses** of each other.

Thus, the answer is: **No, they are not inverses.**
They are not inverses.

Are the following functions inverses? \( \begin{array}{l}f(x)=4 x-2 \\ g(x)=\frac{x}{4}+2 \\ \text { No, they are not inverses } \\ \text { Yes, they are inverses }\end{array} \)

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