Problem 42 - College Algebra Essentials

Question

Evaluate each piece wise function at the given values of the independent variable. \(h(x)=\left\\{\begin{array}{cc}\frac{x^{2}-25}{x-5} & \text { if } x \neq 5 \\\ 10 & \text { if } x=5\end{array}\right.\) a. \(h(7)\) b. \(h(0)\) c. \(h(5)\)

Step-by-Step Solution

Answer

The values of the piecewise function h(x) at the given points are: h(7) = 12, h(0) = 5, and h(5) = 10

Step 1: Analyze the Function at x=7

Since 7 is not equal to 5, the rule for h(x) is \(\frac{x^{2}-25}{x-5}\). So we substitute x=7 into the function to compute h(7) which results into h(7) = \(\frac{7^{2}-25}{7-5} = \frac{49-25}{2} = 12\)

Step 2: Analyze the Function at x=0

Since 0 is not equal to 5, the rule for h(x) is \(\frac{x^{2}-25}{x-5}\). So we substitute x=0 into the function to compute h(0) which results into h(0) = \(\frac{0^{2}-25}{0-5} = \frac{-25}{-5} = 5\)

Step 3: Analyze the Function at x=5

Here x is equal to 5, so the rule for h(x) is 10. Therefore, h(5) = 10

Key Concepts

Evaluating Functions

When we talk about evaluating functions, we are referring to the process of finding the value of a function for a specific input. In the example of the piecewise function h(x), evaluating it at different values of x involves plugging those values into the correct expression defined by the function.

In the given exercise, evaluating h(x) at x=7 and x=0 required us to use the expression \(\frac{x^{2}-25}{x-5}\) since the condition x ≠ 5 is met. For x=5, however, the function dictated we use the constant value 10, illustrating how piecewise functions can change rules based on the input. This approach to function evaluation is foundational in understanding complex algebraic structures.

Algebraic Functions

Algebraic functions are expressions that use algebraic operations, such as addition, subtraction, multiplication, division, and raising to powers, to relate an input to an output. The piecewise function h(x) is algebraic, with two distinct parts. The first rule \(\frac{x^{2}-25}{x-5}\) involves a quadratic expression \(x^{2}\), subtraction, and division by \(x-5\), except for when x equals 5. At this point, to avoid division by zero, a different, simple algebraic function takes over, assigning a constant value.

This demonstrates that algebraic functions can tackle complex scenarios by breaking them into simpler, piecewise segments, easing the process of evaluation and providing a meaningful outcome across their domain.

Function Notation

Function notation is a concise way to represent functions in mathematics, writing them in the form of f(x), where f denotes the function, and x represents the input variable or argument. It's not just symbolic; it is a powerful way to delineate the relationship between variables.

For our piecewise function h(x), the notation changes based on the condition of x. If x is equal to 5, the notation h(5) directly equals 10. It's clear, direct, and avoids ambiguity, enabling us to communicate complex mathematical ideas efficiently. Mastering this notation is essential for any student diving into the world of algebra and higher mathematics.