Problem 18 - 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. $$g(3)$$

Step-by-Step Solution

Answer

Given the function $$ g(t) = t^2 - 1 $$, determine the value of g(3). Answer: 8

Step 1: Write the equation for g(t)

First, let's write down the equation for the function g(t): $$ g(t) = t^2 - 1 $$

Step 2: Substitute t with 3

Now, we'll substitute the value of t = 3 into the equation: $$ g(3) = (3)^2 - 1 $$

Step 3: Evaluate the expression

Evaluating the expression to compute the value of g(3): $$ g(3) = (9) - 1 $$

Step 4: Simplify the result

Lastly, simplify the result to find the value of g(3): $$ g(3) = 8 $$ So, the value of the function g(t) at t = 3 is 8.

Key Concepts

Functions

A function is a mathematical relationship where each input value is paired with exactly one output value. It is often represented as \( f(x) \), \( g(t) \), or \( h(x) \), indicating different functions that operate on different variables. Functions can be visualized as machines, where you input a number, and the machine processes it and gives out a result. In the exercise we are dealing with, the function \( g(t) = t^2 - 1 \) is given. This means each input, \( t \), is squared and then decreased by one to produce an output. Understanding functions involves knowing how to identify the input and apply the relevant operations, which are defined by the function's equation.

Substitution

Substitution is a fundamental technique used to evaluate functions at specific points. It involves replacing the variable in a function's expression with a given number. To effectively perform substitution, follow these steps:

• Identify the variable in the function. For \( g(t) \), the variable is \( t \).
• Determine the value to substitute for this variable. In the exercise, this value is 3.
• Replace every occurrence of the variable in the function's expression with the given number. For \( g(t) = t^2 - 1 \), substitute \( t \) with 3, resulting in \( g(3) = 3^2 - 1 \).

This process makes it possible to transform a function's expression into a numerical expression, which can be evaluated to find the function's value at a particular point.

Algebraic Manipulation

Algebraic manipulation is the process of performing mathematical operations to simplify expressions or solve equations. It requires applying the rules of arithmetic systematically. Here are the key steps used in the exercise:

• Square the substituted value. In our example, this would mean taking \( 3^2 = 9 \).
• Complete any additional operations defined by the function. For \( g(t) = t^2 - 1 \), the next step after squaring \( t \) was to subtract 1 from the result, leading to \( 9 - 1 \).
• Simplify the numerical expression to find the final answer. This resulted in \( g(3) = 8 \).

Understanding algebraic manipulation is crucial for evaluating functions accurately, as it ensures that you can logically and systematically break down and compute expressions.