Problem 101 - Algebra for College Students

Question

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$f(-1) \cdot h(-4)$$

Step-by-Step Solution

Answer

The result is -10.

Step 1: Evaluate the function f at x = -1

First, substitute -1 for x in the function f. We have: \[f(x) = 3x - 2\]So, \[f(-1) = 3(-1) - 2 = -3 - 2 = -5\]

Step 2: Evaluate the function h at x = -4

Next, substitute -4 for x in the function h. We have: \[h(x) = |x + 2|\]So, \[h(-4) = |-4 + 2| = |-2| = 2\]

Step 3: Multiply f(-1) and h(-4)

Now multiply the results from Step 1 and Step 2: \[f(-1) \times h(-4) = -5 \times 2 = -10\]

Key Concepts

Function Notation

In mathematics, function notation is a simple way to represent and work with functions. For a given function, say \( f \), the notation \( f(x) \) denotes the value of the function at some input \( x \). It provides a way to plug in different values into the function and calculate corresponding outputs.
For example, if we have a function \( f(x) = 3x - 2 \), and we want to find the value of this function when \( x = -1 \), we write \( f(-1) \). The execution involves substituting \( -1 \) into the function:

• \( f(-1) = 3(-1) - 2 = -3 - 2 = -5 \)

This notation makes it very clear and efficient to communicate specific evaluations of functions.

Absolute Value Function

An absolute value function is a function that contains an algebraic expression within absolute value symbols, denoted as \(|x|\).
The absolute value of a number is its distance from zero on the number line, regardless of direction, making it always non-negative.
Mathematically:

• For any real number \(a\), \(|a| = a\) if \(a \geq 0 \)
• For any real number \(a\), \(|a| = -a\) if \(a < 0 \)

For example, if we have \( h(x) = |x + 2| \) and we need to evaluate it at \( x = -4 \),

• First, substitute \(-4\) for \(x\): \( h(-4) = |-4 + 2| = |-2| \)
• Then, apply the absolute value operation to \(-2\): \(|-2| = 2 \)

The result is \( h(-4) = 2 \). The absolute value ensures we always get a non-negative output.

Multiplication of Functions

The multiplication of functions involves taking the product of the values of two functions, evaluated at the same or different points.
When presented with an expression such as \( f(x) \times g(x) \), you first evaluate each function individually at the given input or inputs and then multiply the results.
For instance, the given question requires evaluation of \( f(-1) \) and \( h(-4) \), multiplying these results together:

• First, calculate \( f(-1) \): \( f(-1) = -5 \)
• Next, calculate \( h(-4) \): \( h(-4) = 2 \)
• Finally, multiply these values: \( f(-1) \times h(-4) = -5 \times 2 = -10 \)

Through these straightforward steps, you can easily manage the multiplication of function values.