Problem 100 - 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. $$ \frac{h(-10)}{f(2)} $$

Step-by-Step Solution

Answer

2

Step 1: Evaluate h(-10)

Use the definition of the function \(h(x) = |x + 2|\) to evaluate \(h(-10)\). Substitute -10 for x: \(h(-10) = |-10 + 2|\). Simplify inside the absolute value: \(h(-10) = |-8|\). The absolute value of -8 is 8: \(h(-10) = 8\).

Step 2: Evaluate f(2)

Use the definition of the function \(f(x) = 3x - 2\) to evaluate \(f(2)\). Substitute 2 for x: \(f(2) = 3(2) - 2\). Simplify the expression: \(f(2) = 6 - 2\). Subtract 2 from 6: \(f(2) = 4\).

Step 3: Compute \(\frac{h(-10)}{f(2)}\)

Now substitute the values obtained in Steps 1 and 2 into the given expression: \(\frac{h(-10)}{f(2)} = \frac{8}{4}\). Simplify the fraction: \(\frac{8}{4} = 2\).

Key Concepts

Absolute Value Function

The absolute value function requires us to find the distance of a number from zero on the number line. It is always positive or zero. The absolute value of a number is denoted by vertical bars around the number, like this: \(|x|\). For example, \(|-8| = 8\) and \(|8| = 8\). In our example, we evaluated \(|-10 + 2| = |-8| = 8\). This means h(x) when x = -10 is 8.

• Always remember that the absolute value function turns any negative input into a positive output.
• It helps simplify problems involving distances and magnitudes.
• For any real number x, \(|x| = x\) if x is non-negative and \(|x| = -x\) if x is negative.

Linear Function

A linear function is any function that can be written in the form \(y = mx + b\), where m and b are constants. This type of function graphs to a straight line. The rate of change is constant, and the graph's slope is determined by m. In the problem, we used the linear function \(f(x) = 3x - 2\).

When evaluating \(f(2)\), we substituted 2 for x and computed: \(3(2) - 2 = 6 - 2 = 4\). Linear functions are straightforward as each change in x results in a constant change in y.

• The x-coefficient (m) determines the slope of the line.
• The y-intercept (b) is where the line crosses the y-axis.
• They make calculations simpler because the relationship between x and y values remains consistent.

Function Simplification

Function simplification is about making complex expressions easier to handle. In this exercise, once we evaluated \(h(-10)\) and \(f(2)\), we simplified our final expression: \(\frac{h(-10)}{f(2)} = \frac{8}{4}\). This equals 2.

Simplification often involves reducing fractions, combining like terms, and applying basic arithmetic operations. Here are some tips for simplification:

• Reduce fractions to their simplest form by dividing the numerator and the denominator by their greatest common divisor (GCD).
• Combine like terms, which are terms with the same variables raised to the same power.
• Always perform operations inside parentheses first, following the order of operations (PEMDAS/BODMAS).

By simplifying the expressions, we make it much easier to understand and interpret the results. In this case, it led us to the clear and simple answer of 2.