If a function $$ f(x) = 3x + 2 $$, what is its inverse function?

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UpStudy AI · Accepted Answer

To find the inverse of the function $$ f(x) = 3x + 2 $$, follow these steps:

### Step 1: Replace $$ f(x) $$ with $$ y $$
$$
y = 3x + 2
$$

### Step 2: Swap $$ x $$ and $$ y $$
$$
x = 3y + 2
$$

### Step 3: Solve for $$ y $$
$$
x - 2 = 3y
$$
$$
y = \frac{x - 2}{3}
$$

### Step 4: Replace $$ y $$ with $$ f^{-1}(x) $$
$$
f^{-1}(x) = \frac{x - 2}{3}
$$

### Final Answer
The inverse function of $$ f(x) = 3x + 2 $$ is:
$$
f^{-1}(x) = \frac{x - 2}{3}
$$

### Verification
To ensure that $$ f^{-1}(x) $$ is indeed the inverse of $$ f(x) $$, we can check if:
$$
f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x
$$

1. **Check $$ f(f^{-1}(x)) $$:**
$$
f\left( \frac{x - 2}{3} \right) = 3 \left( \frac{x - 2}{3} \right) + 2 = (x - 2) + 2 = x
$$

2. **Check $$ f^{-1}(f(x)) $$:**
$$
f^{-1}(3x + 2) = \frac{(3x + 2) - 2}{3} = \frac{3x}{3} = x
$$

Both compositions result in $$ x $$, confirming that $$ f^{-1}(x) = \frac{x - 2}{3} $$ is indeed the inverse function of $$ f(x) = 3x + 2 $$.
The inverse function of $$ f(x) = 3x + 2 $$ is $$ f^{-1}(x) = \frac{x - 2}{3} $$.

If a function , what is its inverse function?

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