Problem 48 - Precalculus with Limits

Question

In Exercises 37-52, evaluate the function at each specified value of the independent variable and simplify. \(f(x) = |x|+4\) (a) \(f(2)\) (b) \(f(-2)\) (c) \(f(x^2)\)

Step-by-Step Solution

Answer

The answers are: (a) \(f(2) = 6\), (b) \(f(-2) = 6\), and (c) \(f(x^2) = x^2 + 4\).

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

Substitute \(x = 2\) into the function. So, \(f(2) = |2|+4 = 2 + 4 = 6\)

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

Substitute \(x = -2\) into the function. The absolute value of a negative number turns it into its positive counterpart, hence \(|-2| = 2\). So, \(f(-2) = |-2|+4 = 2 + 4 = 6\)

Step 3: Evaluate \(f(x^2)\)

Substitute \(x^2\) for \(x\) in the function. Here we need to remember that the square of any number (negative or positive) is always positive. This means that \(|x^2| = x^2\). Thus, \(f(x^2) = |x^2|+4 = x^2 + 4\)

Key Concepts

Function Evaluation

In mathematics, function evaluation is the process of finding the output of a function for specific input values. Think of a function as a machine that takes an input, performs some operation, and outputs a result. This output is obtained by substituting the input value into the function's equation and simplifying the expression.
To illustrate, consider the function \[ f(x) = |x| + 4 \]. When evaluating this function, if you're given a value like 2 or -2, you replace the variable \( x \) with that number and perform the operations specified by the function. For example:

• For \( f(2) \), substitute 2 for \( x \) to get \( f(2) = |2| + 4 = 2 + 4 = 6 \).
• For \( f(-2) \), substitute -2 for \( x \) to find \( f(-2) = |-2| + 4 = 2 + 4 = 6 \).

By understanding these steps, you can accurately evaluate functions for any given input.

Absolute Value

The absolute value is a crucial concept in mathematics, representing the distance between a number and zero on the number line. It is always non-negative, meaning it is either zero or positive. The absolute value of a number \( x \) is denoted by \( |x| \).
Here are some key points to remember:

• If \( x \) is positive or zero, then \( |x| = x \).
• If \( x \) is negative, then \( |x| = -x \). This effectively changes the sign of \( x \) to positive.

For instance, when evaluating \( f(x) = |x| + 4 \), the absolute value is used to ensure that the output is always on the positive side of the number line regardless of whether \( x \) is negative or positive. This is especially important when evaluating \( f(-2) \), where we change \( |-2| \) to \( 2 \), leading us to the result \( 6 \).
The absolute value function helps flatten out negative inputs, making them positive, ensuring consistency in function evaluations.

Operations with Functions

Operations with functions often involve performing algebraic operations like addition, subtraction, multiplication, or division on one or more functions. Such operations are fundamental when applying specific rules or transformations on functions.
When evaluating functions, particularly when substituting complex expressions such as \( x^2 \) into a function like \( f(x) = |x| + 4 \), it is important to consider the properties of what you're substituting:

• When \( x^2 \) is substituted, observe \( f(x^2) = |x^2| + 4 \). Since any square, whether from a negative or positive number, results in a non-negative number, \( |x^2| = x^2 \).

After substitution, it becomes \( f(x^2) = x^2 + 4 \), maintaining the concept of absolute values turning any input into a non-negative output. Such operations allow us to analyze how functions behave under different manipulations and ensure accurate outputs.
Understanding these foundational operations helps streamline solving function-related problems efficiently.