Problem 15 - Contemporary Precalculus

Question

Refer to these three functions: $$ \begin{aligned} f(x) &=\sqrt{x+3}-x+1 \\ g(t) &=t^{2}-1 \\ h(x) &=x^{2}+\frac{1}{x}+2 \end{aligned} $$ In each case, find the indicated value of the function. $$h(-x)$$

Step-by-Step Solution

Answer

Answer: The expression for \(h(-x)\) is \(h(-x) = x^2 - \frac{1}{x} + 2\).

Step 1: We are given the function \(h(x) = x^2 + \frac{1}{x} + 2\). #Step 2: Substitute \(-x\) for \(x\) in the given function#

We need to find \(h(-x)\). So, we will substitute \(-x\) for \(x\) in the given function: \(h(-x) = (-x)^2 + \frac{1}{-x} + 2\). #Step 3: Simplify the expression for \(h(-x)\)#

Step 2: Now, let's simplify the expression: $$ \begin{aligned} h(-x) &= (-x)^2 + \frac{1}{-x} + 2 \\ &= (-1)^2 x^2 + \frac{1}{-1x} + 2 \\ &= x^2 - \frac{1}{x} + 2 \end{aligned} $$ #Step 4: State the final expression for \(h(-x)\)#

The final expression for \(h(-x)\) is \(h(-x) = x^2 - \frac{1}{x} + 2\).

Key Concepts

Precalculus Fundamentals

Precalculus is a course that prepares students for calculus but it is much more than just a prerequisite. It encompasses a variety of topics that are foundational to higher-level mathematics, including functions and their transformations.

Understanding how functions work and how to manipulate them forms the basis for many concepts in both precalculus and calculus. Topics typically covered include polynomial, rational, exponential, logarithmic, and trigonometric functions, as well as complex numbers and vectors.

When working with functions in precalculus, it's critical to understand their properties, such as domain and range, and how different operations affect them. This knowledge is fundamental for tackling calculus problems later on.

Function Evaluation

Function evaluation is a pivotal skill in mathematics, which involves finding the output of a function given an input. The process often requires substituting the input variable with a given number or expression, as we did with h(-x) in the original problem.

Here are a few key points to remember when evaluating functions:

• Always replace the input variable with the provided value or expression.
• Pay close attention to operation signs; they can alter the output dramatically.
• If the input is an expression, like -x, ensure that you apply the operations to the entire expression.

Properly evaluating functions allows us to analyze and understand the behavior of functions in different scenarios.

Function Simplification

The simplification of functions makes them easier to analyze and work with. When simplifying, we aim to write the function in the most compact form possible while maintaining its original value.

To simplify functions, you often combine like terms, factor expressions, and cancel common factors. It's also crucial to be aware of the rules of exponents and to simplify fractions when applicable. In our example, squaring -x yields x^2, and we need to keep in mind that dividing by negative one inverts the sign.

Common Mistakes to Avoid:

• Not applying operations to all terms within parentheses.
• Forgetting to simplify negative exponents or complex fractions.

With practice, function simplification becomes an intuitive process, allowing for a more effective evaluation of functions.