Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-10,\) find each of the function values when \(\mathrm{f}(x)=4 x\) $$ \mathrm{f}\left(\mathrm{f}^{-1}(\sqrt{2})\right) $$

Step-by-Step Solution

Answer

The value is \( \sqrt{2} \).

Step 1: Identify the Inverse Function

To find the value of \( f(f^{-1}(\sqrt{2})) \), we first need the inverse of the function \( f(x) = 4x \). The inverse function \( f^{-1}(x) \) is determined by solving for \( x \) in terms of \( y \) from the equation \( y = 4x \).

Step 2: Solve for Inverse Function

From \( y = 4x \), we find the inverse function \( f^{-1}(x) \) by rearranging to get \( x = \frac{y}{4} \). Hence, \( f^{-1}(x) = \frac{x}{4} \).

Step 3: Apply Inverse Function

Now, apply the inverse function to \( \sqrt{2} \):\[ f^{-1}(\sqrt{2}) = \frac{\sqrt{2}}{4} \].

Step 4: Evaluate the Original Function

Use \( f(x) \) on the previous result: evaluate \( f\left(\frac{\sqrt{2}}{4}\right) = 4 \times \frac{\sqrt{2}}{4} \).

Step 5: Simplify the Expression

Simplify \( 4 \times \frac{\sqrt{2}}{4} \) to get \( \sqrt{2} \).

Key Concepts

Function Evaluation

Function Evaluation is essential when working with functions as it involves calculating the output for a given input value. In our original exercise, the function is defined as \( f(x) = 4x \). The task is to evaluate \( f(f^{-1}(\sqrt{2})) \).

* The first step involves understanding how to apply an inverse function to a particular value. We start by identifying the inverse of \( f(x) \).

* Next, we apply this inverse function to a specified input, which in this case is \( \sqrt{2} \).

* Finally, evaluate the original function \( f(x) \) using this result. By performing these calculations, you determine the image of \( \sqrt{2} \) under the function \( f \).

Function evaluation is fundamental in gaining insights into a function's behavior. It helps relate specific input values to their corresponding outputs.

Simplifying Expressions

Simplifying expressions is a crucial step in solving mathematical problems. It involves reducing an expression to its most basic form, making it easier to work with. In our solution, this concept was highlighted in step 5.

Here's a breakdown:

* We started with the initial expression \( 4 \times \frac{\sqrt{2}}{4} \). New Insights: Identifying a common factor helps in decreasing the complexity early on.

* By noticing that the \( 4 \) in the numerator cancels with the \( 4 \) in the denominator, the expression simplifies to \( \sqrt{2} \).

In mathematics, simplifying is not just about getting the answer quickly; it's about efficiency and clarity. It also avoids unnecessary complications that might arise from more convoluted expressions. Simplification aids comprehension and corrects interpretations of results.

Algebraic Manipulation

Algebraic Manipulation entails rearranging equations and expressions to solve for variables or simplify the problem. It combines a set of rules and techniques to transform expressions. In this exercise, we dealt with manipulating to find an inverse function.

Here's how we applied it:

• The equation \( y = 4x \) was transformed to solve for \( x \).
  
  This required isolating \( x \) which led to \( x = \frac{y}{4} \).
• The inverse function is the result of such algebraic manipulation, given by \( f^{-1}(x) = \frac{x}{4} \).
  
  

Algebraic manipulation is fundamental in deriving inverse functions and other variables. It provides tools needed to structure and modify equations, so that they deliver required results. By mastering these techniques, finding inverse functions and solving various algebraic problems becomes much more manageable.