Problem 57 - Precalculus Essentials

Question

Evaluate each piece wise function at the given values of the independent variable. \(h(x)=\left\\{\begin{array}{ccc}\frac{x^{2}-9}{x-3} & \text { if } & x \neq 3 \\\ 6 & \text { if } & x=3\end{array}\right.\) a. \(h(5)\) b. \(h(0)\) c. \(h(3)\)

Step-by-Step Solution

Answer

Based on the steps and calculations, the answers are \( h(5) = 8 \), \( h(0) = 3 \), and \( h(3) = 6 \).

Step 1: Evaluation of \( h(5) \)

The value 5 is not equal to 3, so the first part of the piecewise function applies. The expression should be evaluated as \( h(5) = \frac{5^2 - 9}{5 - 3} = \frac{16}{2} = 8. \)

Step 2: Evaluation of \( h(0) \)

The value 0 also does not equal to 3, so again, the first part of the piecewise function should be used. This gives \( h(0) = \frac{0^2 - 9}{0 - 3} = \frac{-9}{-3} = 3. \)

Step 3: Evaluation of \( h(3) \)

In this case, the value 3 equals 3. Therefore, the second part of the piecewise function applies. This means \( h(3) \) equals to 6.

Key Concepts

Function Evaluation

Function evaluation involves determining the output of a function for a given input value, which is the independent variable. In simpler terms, if you have a function, you want to know what result you get when you plug a specific value into it. In the case of piecewise functions, this can be a bit tricky because the function is defined by different expressions depending on the input value.

• For example, look at the function \( h(x) \) provided in the exercise. It's composed of two different expressions: \( \frac{x^2 - 9}{x - 3} \) and a constant \( 6 \).
• The expression to use depends on whether \( x \) (the input or independent variable) equals 3 or not.

When evaluating, check the input value against the condition stated in the piecewise function to decide which expression to use. This method is straightforward, but it requires careful attention to the conditions set in the function definition. By following these conditions strictly, you ensure that you evaluate the correct part of the function based on the input value.

Independent Variable

The independent variable is the input or the variable we choose values for in a function. In the function \( h(x) \), \( x \) is the independent variable. You provide different values to \( x \), and the function returns a result for each of these inputs, known as the dependent variable.

• In mathematical terms, if the function is expressed as \( h(x) \), then \( x \) is the independent variable.
• Each function may have specific rules or conditions for different values of the independent variable, as seen in piecewise functions.

Understanding the role of the independent variable is crucial because it dictates which part of a piecewise function you will use. By analyzing the independent variable against the conditions stated in the piecewise function, you reliably evaluate the function. Keep in mind, the evaluation reflects the behavior of the function with respect to different inputs, giving a complete picture through various values of the independent variable.

Precalculus Problem Solving

Precalculus problem solving often involves dealing with different types of functions, including piecewise functions. This requires an understanding of how to approach evaluating different conditions or parts of a function based on its definition and the values of the independent variable.

• The exercise is an excellent example of this, demonstrating the evaluation of a piecewise function through several distinct scenarios.
• You need to match the value of the independent variable with the appropriate condition or equation described in the piecewise function.

Problem-solving in precalculus involves systematic thinking and logical deductions. To tackle piecewise functions efficiently, practice organizing your steps:

• Identify the function's expression that matches your input condition.
• Substitute the input value into the correct expression and solve for the output.

Through consistent practice, your problem-solving skills grow, making it easier to handle complex functions and equations as you progress in mathematics.