Let $$ f(x) = 5x + 1 $$. Determine its inverse function and demonstrate that they are inverses by showing that both compositions $$ f(f^{-1}(x)) $$ and $$ f^{-1}(f(x)) $$ return x.

Question

UpStudy AI · Accepted Answer

To find the inverse of the function $$ f(x) = 5x + 1 $$ and demonstrate that they are indeed inverses, follow these steps:

### 1. Finding the Inverse Function $$ f^{-1}(x) $$

To find the inverse function, we need to solve the equation $$ y = 5x + 1 $$ for $$ x $$ in terms of $$ y $$:

$$
y = 5x + 1
$$

Subtract 1 from both sides:

$$
y - 1 = 5x
$$

Now, divide both sides by 5 to solve for $$ x $$:

$$
x = \frac{y - 1}{5}
$$

Therefore, the inverse function is:

$$
f^{-1}(x) = \frac{x - 1}{5}
$$

### 2. Verifying the Inverses through Composition

To confirm that $$ f $$ and $$ f^{-1} $$ are indeed inverses, we need to show that:

1. $$ f(f^{-1}(x)) = x $$
2. $$ f^{-1}(f(x)) = x $$

#### a. Composition $$ f(f^{-1}(x)) $$

$$
f(f^{-1}(x)) = f\left(\frac{x - 1}{5}\right) = 5\left(\frac{x - 1}{5}\right) + 1
$$

Simplify the expression:

$$
5 \times \frac{x - 1}{5} = x - 1
$$

$$
x - 1 + 1 = x
$$

So,

$$
f(f^{-1}(x)) = x
$$

#### b. Composition $$ f^{-1}(f(x)) $$

$$
f^{-1}(f(x)) = f^{-1}(5x + 1) = \frac{(5x + 1) - 1}{5}
$$

Simplify the expression:

$$
\frac{5x + 1 - 1}{5} = \frac{5x}{5} = x
$$

So,

$$
f^{-1}(f(x)) = x
$$

### Conclusion

Since both compositions $$ f(f^{-1}(x)) $$ and $$ f^{-1}(f(x)) $$ return $$ x $$, we have confirmed that $$ f(x) = 5x + 1 $$ and its inverse $$ f^{-1}(x) = \frac{x - 1}{5} $$ are indeed inverse functions of each other.

**Final Answer:**

The inverse function is $$ f^{-1}(x) = \dfrac{x - 1}{5} $$. Demonstrating the compositions:

$$
f(f^{-1}(x)) = f\left(\dfrac{x - 1}{5}\right) = 5 \cdot \dfrac{x - 1}{5} + 1 = x - 1 + 1 = x
$$
$$
f^{-1}(f(x)) = f^{-1}(5x + 1) = \dfrac{5x + 1 - 1}{5} = \dfrac{5x}{5} = x
$$

Thus, $$ f $$ and $$ f^{-1} $$ are inverses since both compositions equal $$ x $$.
The inverse function of $$ f(x) = 5x + 1 $$ is $$ f^{-1}(x) = \frac{x - 1}{5} $$. Both compositions $$ f(f^{-1}(x)) $$ and $$ f^{-1}(f(x)) $$ equal $$ x $$, confirming that they are inverses of each other.

Let . Determine its inverse function and demonstrate that they are inverses by showing that both compositions and return x.

Upstudy AI Solution

Answer

Solution

1. Finding the Inverse Function

2. Verifying the Inverses through Composition

a. Composition

b. Composition

Conclusion

Beyond the Answer

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