Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-10,\) find each of the function values when \(\mathrm{f}(x)=4 x\) $$ \mathrm{f}\left(\mathrm{f}^{-1}(-6)\right) $$

Step-by-Step Solution

Answer

The value of \( f(f^{-1}(-6)) \) is \(-6\).

Step 1: Identify the Inverse Function

To find the inverse of the function \( f(x) = 4x \), we need to express \( x \) in terms of \( y \). Start by writing \( y = 4x \), then solve for \( x \).

Step 2: Solve for the Inverse Function

From \( y = 4x \), solve for \( x \) by dividing both sides by 4: \( x = \frac{y}{4} \). Thus, \( f^{-1}(x) = \frac{x}{4} \).

Step 3: Evaluate the Inverse Function at -6

Plug \(-6\) into the inverse function we found: \( f^{-1}(-6) = \frac{-6}{4} = -\frac{3}{2} \).

Step 4: Substitute back into the Original Function

Now substitute \( f^{-1}(-6) \) into the original function: \( f\left(-\frac{3}{2}\right) = 4(-\frac{3}{2}) \).

Step 5: Calculate the Result

Calculate \( 4(-\frac{3}{2}) \) to get \( -6 \). Therefore, \( f(f^{-1}(-6)) = -6 \).

Key Concepts

Inverse Function

Understanding inverse functions is crucial in mathematics as it allows us to reverse the effect of a function. An inverse function essentially "undoes" what the original function does. For instance, if you have a function \( f(x) = 4x \), it multiplies any input \( x \) by 4. The task of finding the inverse function \( f^{-1}(x) \) is to figure out how to get back the original input \( x \) from the output \( y \).

To do this, we start with the equation \( y = 4x \) and solve for \( x \). This involves algebraic manipulation like reversing operations. You divide both sides by 4, which gives you:

• \( x = \frac{y}{4} \)

Ultimately, this means our inverse function is \( f^{-1}(x) = \frac{x}{4} \), effectively dividing the input by 4.

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. This concept is quite straightforward once the function and its inverse are known. For the given function \( f(x) = 4x \), evaluating \( f \) at any \( x \) just means calculating \( 4 \times x \).

When we previously evaluated the inverse function at \(-6\), we found:

• \( f^{-1}(-6) = \frac{-6}{4} = -\frac{3}{2} \)

Next, we use this result to evaluate the original function at this new input. We substitute \( -\frac{3}{2} \) into \( f(x) \), resulting in:

• \( f\left(-\frac{3}{2}\right) = 4 \times -\frac{3}{2} \)

This results in \( -6 \). Thus, \( f(f^{-1}(-6)) = -6 \), proving our calculations are correct.

Algebraic Manipulation

The ability to manipulate algebraic expressions efficiently is a key skill in solving equations involving functions and their inverses. Algebraic manipulation involves transforming expressions to isolate desired variables or simplify equations. In our solution, we used these skills to find the inverse function by first setting \( y = 4x \), and then:

• Dividing both sides by 4 to isolate \( x \) on one side.
• This gives \( x = \frac{y}{4} \).

Such manipulations are vital in determining inverse functions. They allow us to solve for one variable in terms of others, enabling us to reverse the function operations. Algebraic manipulation also aids in verifying solutions, ensuring that when applying an inverse function back into the original, we attain the expected result.

Understanding and practicing these skills helps enhance problem-solving capabilities and provides foundational support as you encounter more complex functions and mathematical concepts.