Problem 35 - College Algebra 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. \(f(2)\) b. \(f(-2)\) c. \(f(-x)\)

Step-by-Step Solution

Answer

The simplified expressions are: a) \(f(2)=\frac{15}{4}\) , b) \(f(-2)=\frac{15}{4}\), c) \(f(-x)=\frac{4x^{2}-1}{x^{2}}\).

Step 1: Evaluate \(f(2)\)

Substitute \(x=2\) into the function \(f(x)=\frac{4 x^{2}-1}{x^{2}}\). This gives \(f(2)=\frac{4(2^{2})-1}{(2^{2})}\), which simplifies to \(f(2)=\frac{15}{4}\)

Step 2: Evaluate \(f(-2)\)

Substitute \(x=-2\) into the function \(f(x)=\frac{4 x^{2}-1}{x^{2}}\). This gives \(f(-2)=\frac{4(-2^{2})-1}{(-2^{2})}\), which simplifies to \(f(-2)=\frac{15}{4}\)

Step 3: Evaluate \(f(-x)\)

Substitute \(-x\) for \(x\) in \(f(x)=\frac{4 x^{2}-1}{x^{2}}\). This gives \(f(-x)=\frac{4(-x^{2})-1}{(-x^{2})}\), which can be further simplified to \(f(-x)=\frac{4x^{2}-1}{x^{2}}\)

Key Concepts

Understanding Independent Variables

In mathematics, the independent variable is the input of a function, typically represented by symbols like \(x\). It's the value that we have control over, and other values depend on it. By substituting different numbers for the independent variable, we can see how the function's output changes. This exercise involves evaluating a function, \(f(x)=\frac{4x^2-1}{x^2}\), by substituting specific numbers for \(x\).

• In part (a), the independent variable \(x\) takes the value 2, so we substitute \(x = 2\) into the function.
• In part (b), the independent variable is \(-2\), giving us a different expression to evaluate.
• Finally, in part (c), the independent variable becomes \(-x\), showcasing how the function behaves with a negative input as a variable.

Understanding changes in the independent variable helps us learn how each value affects the overall function's behavior.

Simplifying Expressions Made Easy

Simplifying expressions involves reducing them to their most basic and understandable form. In this context, simplifying refers to the process of evaluating the function \(f(x)=\frac{4x^2-1}{x^2}\) to make it easier to interpret results. Whenever we substitute a value into the function for \(x\), we perform arithmetic operations to simplify.This means carrying out exponentiation, multiplication, subtraction, and division wherever applicable. For example:

• After substituting \(x = 2\), simplify all operations inside the fraction: \(\frac{4(2^2)-1}{2^2}\) to get \(\frac{15}{4}\).
• This step-by-step arithmetic not only solves the problem but also showcases the importance of following the order of operations.

Simplifying provides clarity on the outcomes derived from substitution, giving you a straightforward result from potentially complex expressions.

Substitution Method for Function Evaluation

The substitution method is a crucial mathematical technique used to replace variables with numbers or other expressions. By doing this, we calculate specific outputs that help us understand function behavior. In this exercise, substituting involves plugging given values into \(f(x)=\frac{4x^2-1}{x^2}\).To perform substitution:

• Take the expression \(f(x)\) and replace \(x\) with the given number or expression, like 2, -2, or -x.
• Then, follow through with arithmetic operations to simplify.
• For example, for substitution \(f(-x)\), replace every \(x\) with \(-x\) in the formula, which gives \(f(-x)=\frac{4(-x)^2-1}{(-x)^2}\). Notice that it simplifies back to the original expression \(\frac{4x^2-1}{x^2}\), showing symmetry.

Using substitution correctly allows us to evaluate functions and see how different inputs influence the outputs.