Problem 102 - 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. $$h(0) \cdot g(0)$$

Step-by-Step Solution

Answer

\(h(0) \times g(0) = -4\)

Step 1: Evaluate h(0)

Substitute 0 into the function \(h(x)\): \[h(0) = |0 + 2|\] Calculate the value inside the absolute value: \[h(0) = |2|\] Since the absolute value of 2 is 2: \[h(0) = 2\]

Step 2: Evaluate g(0)

Substitute 0 into the function \(g(x)\): \[g(0) = -0^{2} + 3 \times 0 - 2\] Simplify the expression: \[g(0) = 0 + 0 - 2\] \[g(0) = -2\]

Step 3: Multiply h(0) and g(0)

Now multiply the results obtained from the previous steps: \[h(0) \times g(0) = 2 \times (-2)\] Calculate the product: \[h(0) \times g(0) = -4\]

Key Concepts

Understanding Absolute Value

The absolute value of a number represents its distance from zero on a number line. It is always non-negative.
Absolute value is denoted by vertical bars. For example, \(|x|\) translates to the absolute value of \(x\).
This means if \(x = 2\), then \(|2| = 2\), and if \(x = -3\), then \(|-3| = 3\).
This property was crucial in finding \(h(0)\) in the exercise. Plugging \(0\) into \(h(x) = |x+2|\) simplifies to \(|0+2| = |2|\), and since the absolute value of 2 is 2, we get \(h(0) = 2\).
Always remember:

• Absolute values remove the sign of a number.
• They measure distance, not direction.

Examining Polynomial Functions

Polynomial functions are expressions involving variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents.
For example, the function \(g(x) = -x^2 + 3x - 2\) is a quadratic polynomial. Each term in this polynomial:

• \(-x^2\): A squared term with a coefficient of \(-1\).
• \(3x\): A linear term with a coefficient of 3.
• \(-2\): A constant term.

Evaluating \(g(x)\) for a particular value means substituting that value and performing arithmetic operations. For \(g(0)\), substituting \(0\) yields: \(-0^2 + 3(0) - 2 = -2\). Polynomial functions can take various forms, and understanding each component helps in solving them step by step.

Multiplying Functions

Function multiplication involves multiplying the values of two functions. In our exercise, we multiplied the results from \(h(0)\) and \(g(0)\).
The steps are straightforward:

• First, evaluate each function separately at the given input.
• Next, multiply these evaluated results.

In this case: \(h(0) = 2\) and \(g(0) = -2\), so:
\[ h(0) \times g(0) = 2 \times (-2) = -4\] Function multiplication can be applied broadly in calculations involving multiple functions, by ensuring you correctly evaluate each function before multiplying the outcomes. This provides the final result after steps of individual calculations.