Problem 60 - Elementary and Intermediate Algebra

Question

Find \(h(5)\) and \(h(-2) .\) See Example 4. $$ h(x)=|x|-5 $$

Step-by-Step Solution

Answer

\( h(5) = 0 \) and \( h(-2) = -3 \).

Step 1: Understand the function

The function given is \( h(x) = |x| - 5 \). This is an absolute value function, which means \( |x| \) represents the non-negative value of \( x \). You need to substitute specific values into this function to find \( h(5) \) and \( h(-2) \).

Step 2: Substitute into the function for \( h(5) \)

To find \( h(5) \), substitute \( x = 5 \) into the function. Thus, the expression becomes \( h(5) = |5| - 5 \).

Step 3: Evaluate the expression for \( h(5) \)

Calculate the expression from step 2: \( |5| = 5 \), so \( h(5) = 5 - 5 = 0 \). Hence, \( h(5) = 0 \).

Step 4: Substitute into the function for \( h(-2) \)

To find \( h(-2) \), substitute \( x = -2 \) into the function. The expression is \( h(-2) = |-2| - 5 \).

Step 5: Evaluate the expression for \( h(-2) \)

Calculate the expression from step 4: \( |-2| = 2 \), so \( h(-2) = 2 - 5 = -3 \). Hence, \( h(-2) = -3 \).

Key Concepts

Understanding Evaluating Functions

Evaluating functions is a crucial concept in mathematics that involves finding the value of a function for a specific input. When you are asked to find values like \(h(5)\) or \(h(-2)\), you're essentially plugging these values into the function equation to see what output is produced. The function we are working with is \(h(x) = |x| - 5\). Here, \(|x|\) denotes the absolute value of \(x\), which is always a non-negative number.
To evaluate a function:

• Identify the variable in the function, usually noted by \(x\).
• Replace \(x\) with the given number.
• Solve the mathematical expression after substitution to find the function's output.

This process uses a straightforward arithmetic approach but relies on understanding both the function itself and the mathematical operations involved.

Exploring the Substitution Method

The substitution method is just that—substituting numbers in place of variables in order to solve mathematical expressions and evaluate functions. It's a fundamental technique you will use to extract specific outcomes from general formulas. In our example, you have the function \(h(x) = |x| - 5\). To know what \(h(5)\) equals, you replace \(x\) with 5, leading to: \(h(5) = |5| - 5\). Similarly, to find what \(h(-2)\) equals, replace \(x\) with -2, yielding: \(h(-2) = |-2| - 5\).
Points to remember during substitution:

• Make sure you correctly replace every instance of the variable \(x\) with the given number.
• Handle the absolute value appropriately; remember, the absolute value of a number is always non-negative.
• Perform arithmetic operations accurately after substitution to get the correct result.

The substitution method simplifies complex expressions, allowing you to deal with real numbers instead of variables.

Demystifying Mathematical Expressions

Mathematical expressions involve numbers, symbols, and operators (like addition or multiplication) arranged in a meaningful way. To solve these expressions accurately, you need to understand the rules that govern operations, especially when it involves things like absolute values or subtraction.
The expression \(h(x) = |x| - 5\) tells us that to find \(h(5)\) or \(h(-2)\), you must follow certain rules:

• The absolute value, \(|x|\), always yields a non-negative result. For instance, \(|5| = 5\) and \(|-2| = 2\).
• After obtaining the absolute value, subtract 5 from it to get the final result.

Understanding each component of an expression ensures you approach each step with clarity, allowing you to solve it correctly. This insight is essential, especially when working with more complex or unfamiliar expressions in mathematics.