Problem 21 - 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. $$g(-t)$$

Step-by-Step Solution

Answer

Answer: g(-t) = t^2 - 1

Step 1: Substitute -t for t

Replace the variable t with -t in the given function: $$ g(-t)=(-t)^{2}-1 $$

Step 2: Simplify the expression

Simplify by following the order of operations (PEMDAS) and using the rule for (-t)^2: $$ g(-t)=(-t)^{2}-1=(t^2)-1 $$ So the function $$g(-t) = t^{2}-1$$.

Key Concepts

Function Evaluation

Understanding function evaluation is a fundamental concept in precalculus, which allows you to determine the output of a function for a specific input. In essence, when you evaluate a function, you're replacing each occurrence of the variable with the given number or expression and then simplifying.

Let's consider an example with the function
\( g(t) = t^2 - 1 \)
Suppose you want to evaluate \( g(-t) \). What you do is substitute \( -t \) in for every \( t \) in the function, so \( g(-t) = (-t)^2 - 1 \). Remember that when you substitute in, it's crucial to use parentheses to maintain the integrity of the original expression.
After substitution, simplify the expression using algebraic rules, such as \( (-t)^2 = t^2 \) since squaring a negative number results in a positive. That means \( g(-t) = t^2 - 1 \), which is our evaluated function.

Function Operations

When you're combining, adding, subtracting, multiplying, or dividing functions, this set of procedures is referred to as function operations. It's important to approach these operations methodically. For instance, if you're asked to find \( (f + g)(x) \), you're looking to add the functions \( f(x) \) and \( g(x) \) for a particular value of \( x \).

Continuing with our functions from the exercise, let's say you want to add \( f(x) \) and \( g(t) \). You would write
\( f(x) + g(t) = \left(\sqrt{x + 3} - x + 1\right)+ (t^2 - 1) \)
Each function is treated as a separate entity, and you'd perform the addition term by term. This highlights a crucial point of function operations: the operations are performed on the outputs of the functions, not the inputs.

Order of Operations

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction), is the hierarchy followed to simplify mathematical expressions or evaluate functions consistently. For function evaluation in precalculus, this rule system ensures that everyone arrives at the same result when calculating.

Using the example from our exercise where we evaluate \( g(-t) \), we apply PEMDAS as follows:

• First tackle the Parentheses by substituting \( -t \) for every \( t \).
• Next, address the Exponents by squaring \( -t \) to get \( t^2 \).
• Since there are no Multiplication or Division operations left, we move on to Addition and Subtraction, resulting in \( g(-t) = t^2 - 1 \).

By consistently applying these rules, you avoid common mistakes and correctly evaluate mathematical expressions.