Problem 35 - Precalculus Essentials

Question

Evaluate each function at the given values of the independent variable and simplify. \(f(x)=\frac{4 x^{2}-1}{x^{2}}\) a. \(f(2)\) \(f(-2) \quad\) c. \(f(-x)\)

Step-by-Step Solution

Answer

The function evaluated at \(x = 2\), and \(x = -2\) is 3.75. And the function \(f(x) = f(-x)\).

Step 1: Evaluate for \(x = 2\)

Substitute \(x = 2\) into the function. So we get \(f(2) = \frac{4*(2)^{2}-1}{(2)^{2}} = \frac{16-1}{4} = \frac{15}{4} = 3.75 \)

Step 2: Evaluate for \(x = -2\)

Substitute \(x = -2\) into the function. So we get \(f(-2) = \frac{4*(-2)^{2}-1}{(-2)^{2}} = \frac{16-1}{4} = \frac{15}{4} = 3.75 \)

Step 3: Evaluate for \(x = -x\)

Substitute \(x = -x\) into the function. This gives us a slightly different function. We get \(f(-x) = \frac{4*(-x)^{2}-1}{(-x)^{2}} = \frac{4x^{2}-1}{x^{2}}\). We can now see that \(f(-x)\) equals to \(f(x)\) (the initial function) as the function is an even function.

Key Concepts

Substitute Values into Expressions

Substituting values into expressions is a fundamental skill in mathematics, especially when evaluating functions. This is where you replace every instance of a variable, such as \( x \), with a specified value, ensuring the function outputs a numerical result.

To substitute effectively, follow these steps:

• Identify the value to substitute. For example, in \( f(x) = \frac{4x^2 - 1}{x^2} \), we might substitute \( x = 2 \).
• Replace every occurrence of the variable \( x \) in the expression with the given value.
• Simplify the resulting expression using arithmetic operations and simplification rules.

By substituting \( x = 2 \), the function becomes \( f(2) = \frac{4(2)^2 - 1}{2^2} \). Solve to find numerical results. Practicing this skill helps solve more complex functions easily.

Rational Functions

A rational function is any function that can be expressed as a ratio of two polynomials. The general form of a rational function is \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not zero.

In the provided exercise, \( f(x) = \frac{4x^2-1}{x^2} \) is a rational function. Understanding these functions involves examining:

• **Numerator and Denominator:** Here, \( P(x) = 4x^2 - 1 \) and \( Q(x) = x^2 \). The numerator determines the zeros and the shape, while the denominator determines the domain.
• **Domain:** Rational functions are undefined wherever the denominator is zero. In this case, \( x^2 = 0 \) means \( x = 0 \) is excluded from the domain.

Recognizing rational functions aids in understanding their behavior and solving for their values by manipulating the numerator and denominator.

Function Symmetry

In mathematics, function symmetry determines how a function behaves when inputs are replaced with their opposites. Symmetries can reveal insightful properties about the function's graph.

There are two primary types of symmetry in functions:

• **Even Functions:** These functions satisfy \( f(-x) = f(x) \). They are symmetric around the y-axis. In our example, substituting \( x \) with \( -x \) in \( f(x) = \frac{4x^2 - 1}{x^2} \) shows \( f(-x) \) equals \( f(x) \). This confirms the function is even.
• **Odd Functions:** These satisfy \( f(-x) = -f(x) \). They are symmetric about the origin.

By determining symmetry, you can learn a lot about the graph of a function without plotting points individually. Symmetry can simplify evaluations and solve equations involving the function more efficiently.