Problem 56 - Intermediate Algebra

Question

If \(f(x)=|3 x-2|\) and \(g(x)=|x|+2\), find \(f(1), f(-1)\), \(g(2)\), and \(g(-3)\).

Step-by-Step Solution

Answer

\(f(1) = 1\), \(f(-1) = 5\), \(g(2) = 4\), \(g(-3) = 5\).

Step 1: Evaluate f(x) for x = 1

First, substitute x with 1 in the expression for \( f(x) = |3x - 2| \). Calculate:\( f(1) = |3(1) - 2| = |3 - 2| = |1| = 1 \). Thus, \( f(1) = 1 \).

Step 2: Evaluate f(x) for x = -1

Substitute x with -1 in the expression for \( f(x) = |3x - 2| \). Calculate:\( f(-1) = |3(-1) - 2| = |-3 - 2| = |-5| = 5 \). Therefore, \( f(-1) = 5 \).

Step 3: Evaluate g(x) for x = 2

Substitute x with 2 in the expression for \( g(x) = |x| + 2 \). Calculate: \( g(2) = |2| + 2 = 2 + 2 = 4 \). Thus, \( g(2) = 4 \).

Step 4: Evaluate g(x) for x = -3

Substitute x with -3 in the expression for \( g(x) = |x| + 2 \). Calculate: \( g(-3) = |-3| + 2 = 3 + 2 = 5 \). Therefore, \( g(-3) = 5 \).

Key Concepts

Evaluating Functions

Evaluating functions means finding the result of a function for a given input. To evaluate a function like \( f(x) = |3x - 2| \), you simply need to plug in the value of \( x \) into the equation and then solve. Functions tell you how to manipulate the input number with specific operations to arrive at the output. Here are some simple steps to follow:

• Substitute the input value (e.g., \( x = 1 \) or \( x = -1 \)) into the function.
• Perform the necessary operations as indicated by the function rule (like multiplication and subtraction).
• Compute the final result, preserving the absolute value, if applicable.

When you substitute \( x = 1 \) into \( f(x) \), you calculate:\[f(1) = |3(1) - 2| = |1| = 1\]Revisit the steps for each new input value. With practice, this process becomes straightforward.

Piecewise Functions

Piecewise functions are special because they involve different expressions or rules in different parts of their domain. Although the function \( f(x) = |3x - 2| \) is not explicitly piecewise, absolute value naturally creates a piecewise nature.For example, when \( 3x - 2 \) is positive, \( f(x) \) simply equals \( 3x - 2 \). When \( 3x - 2 \) is negative, the absolute value sign negates the value making \( f(x) = -(3x - 2) \).Steps to understand piecewise behavior:

• Determine when the expression inside the absolute value is positive or negative.
• Use different expressions based on the sign of the input expression.
  Example: If \( 3x - 2 \geq 0 \), then \( f(x) = 3x - 2 \). If \( 3x - 2 < 0 \), then \( f(x) = -(3x - 2) \).

This enables us to evaluate \( f(x) \) for both negative and positive inputs, ensuring that we always arrive at a positive result due to the absolute value.

Mathematical Operations

When dealing with functions, understanding the basic mathematical operations is essential. In this case, the functions involve absolute value, multiplication, addition, and subtraction.Consider the basic operation steps for evaluating \( g(x) = |x| + 2 \):

• Start by calculating the absolute value, which converts any negative number into its positive counterpart.
• After finding the absolute value, add 2 to it.

For example, to find \( g(2) \), compute:\[g(2) = |2| + 2 = 2 + 2 = 4\]And similarly for a negative input:\[g(-3) = |-3| + 2 = 3 + 2 = 5\]Understanding these operations allows you to easily navigate through tasks involving function evaluations. As you practice more with each operation, you'll increase your proficiency and confidence in handling mathematical functions.