Problem:

Given the function $f(x) = 3x + 5$, find its inverse function $f^{-1}(x)$.

Solution:

To find the inverse function $f^{-1}(x)$, follow these steps:

1. Start with the function equation: $y = 3x + 5$ Here, $y$ represents $f(x)$.

2. Swap $x$ and $y$: This step is crucial because the inverse function reverses the roles of the input and the output. $x = 3y + 5$

3. Solve for $y$: Now, we isolate $y$ to find the inverse: $x - 5 = 3y$ $y = \frac{x - 5}{3}$

4. Replace $y$ with $f^{-1}(x)$: The inverse function is: $f^{-1}(x) = \frac{x - 5}{3}$

Conclusion:

The inverse function of $f(x) = 3x + 5$ is $f^{-1}(x) = \frac{x - 5}{3}$.

Verification:

To verify the correctness of the inverse, we can check the compositions $f(f^{-1}(x))$ and $f^{-1}(f(x))$:

• $f(f^{-1}(x)) = 3\left(\frac{x - 5}{3}\right) + 5 = x$
• $f^{-1}(f(x)) = \frac{(3x + 5) - 5}{3} = x$

Thus, both compositions return $x$, confirming that $f^{-1}(x)$ is correct.

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Would you like further details or have any questions?

Here are 5 related questions to explore:

1. What is the inverse of $f(x) = 2x + 7$?
2. How do you find the inverse of a quadratic function?
3. What is the significance of the inverse function in real-world applications?
4. How can you graphically verify if two functions are inverses of each other?
5. What is the domain and range of an inverse function?

Tip: Always check that a function is one-to-one (bijective) before finding its inverse. If it’s not one-to-one, the inverse doesn't exist over the entire domain!