Problem 93 - 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. $$ h(-3) $$

Step-by-Step Solution

Answer

\( h(-3) = 1 \)

Step 1: Understand the function

The function given is: \[ h(x) = |x + 2| \]This is an absolute value function, which means it measures the distance of the expression inside from zero.

Step 2: Substitute the value of x

We need to evaluate the function at \(x = -3\). Substitute \( -3 \) into the function:\[ h(-3) = |-3 + 2| \]

Step 3: Simplify the expression inside the absolute value

Compute the expression inside the absolute value:\[ h(-3) = |-1| \]

Step 4: Take the absolute value

Now apply the absolute value function. The absolute value of \( -1 \) is \( 1 \):\[ h(-3) = 1 \]

Key Concepts

absolute value

The absolute value is a fundamental concept in mathematics. It tells us the distance of a number from zero on a number line. Absolute values are always positive or zero. This is because distance cannot be negative.

In an equation, if we have \( |x| \), it reads as 'the absolute value of x'. For example:

• \( |5| = 5 \)
• \( |-3| = 3 \)

This shows the absolute value of both positive and negative numbers.

So, when faced with an absolute value function like \(|x + 2| \), the idea is to take whatever is inside the bars, compute its value, and then strip away any negative sign.

function evaluation

Evaluating a function means finding the output for a given input. Functions are like machines: you put something in, and you get something out. If you have a function \ f(x) = 2x + 3 \, and evaluate it at \(x = 4 \), you would compute \ f(4) \ like this:

• Substitute \(4 \) into the function: \ f(4) = 2 \cdot 4 + 3 \
• Multiply and add to find the output: \ f(4) = 8 + 3 = 11 \

Applying this to our absolute value function \ h(x) = |x + 2| \, evaluating \ h(-3) \ involves similar steps.

Plug in \( -3 \) for \( x \): \( h(-3) = |-3 + 2| \).
Then, simplify the inside to get \( |-1| \).

substituting values

Substituting values is a key step in function evaluation. It involves replacing the variable in the function with a specific number.

For the given problem, you have the function \( h(x) = |x + 2|\) and need to find \( h(-3) \). Here's how:

• Start by writing the function: \ h(x) = |x + 2| \
• Substitute \( -3 \) for \( x \): \ h(-3) = |-3 + 2| \
• Simplify inside the absolute value: \ h(-3) = |-1| \

After substituting and simplifying, you get the value inside the absolute value function. Next, compute the absolute value as explained in the earlier section. This gives the final value of \(h(-3) = 1\).

Substitution transforms a general function into a specific number, making it easier to work with.