Problem 39 - Intermediate Algebra

Question

Let \(f(x)=2 x+1\) and \(g(x)=x^{2}-1 .\) Find each of the following. See Example 3 . $$ (g \circ f)(-3) $$

Step-by-Step Solution

Answer

\((g \circ f)(-3) = 24\).

Step 1: Understand the composition of functions

To find \((g \circ f)(x)\), we need to understand that this means we're plugging the output of \(f(x)\) into \(g(x)\). Therefore, \((g \circ f)(x) = g(f(x))\). This is called the composition of functions.

Step 2: Calculate \(f(-3)\)

Since \(f(x) = 2x + 1\), we substitute \(-3\) for \(x\) to find \(f(-3)\).\[f(-3) = 2(-3) + 1 = -6 + 1 = -5\]

Step 3: Substitute \(f(-3)\) into \(g\)

Now, use the result from Step 2 as the input for \(g(x)\). So we need to find \(g(-5)\).Since \(g(x) = x^2 - 1\), substitute \(-5\) for \(x\):\[g(-5) = (-5)^2 - 1 = 25 - 1 = 24\]

Step 4: Review the result

The value of \((g \circ f)(-3)\) is \( g(-5) \), which we calculated as 24.

Key Concepts

Function Evaluation

Function evaluation is the process of finding the value of a function for a specific input. It's like asking, "What is the result when I use this number?" When you have a function, you can input different values into it to see what results you get.
For example, if you have the function \( f(x) = 2x + 1 \), and you want to evaluate it at \(-3\), you just replace every \(x\) in the function with \(-3\). This gives you:

• Start with \( f(x) = 2x + 1 \)
• Substitute \(-3\) for \(x\), giving \( f(-3) = 2(-3) + 1 \)
• Calculate the result: \(-6 + 1 = -5\)

Function evaluation lets you understand how a function works with different numbers. It's a crucial step in solving problems, especially when working with composed functions like \((g \circ f)(x)\). By evaluating each function step by step, you can simplify complex problems.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators. These expressions can represent real-world situations and are essential in writing functions.

• Consider \( f(x) = 2x + 1 \). It has a variable \(x\), a coefficient \(2\), and a constant \(+1\).
• Similarly, \( g(x) = x^2 - 1 \) is another algebraic expression with the variable squared, showing how different algebraic forms can be combined.

When you compose functions, the understanding of algebraic expressions becomes vital. In our exercise, we combined \(f(x)\) and \(g(x)\) by inputting the results of one function into another. Recognizing and simplifying these expressions helps in making calculations easier and more intuitive.

Mathematical Operations

Mathematical operations form the backbone of evaluating functions and interpreting algebraic expressions. Operations include addition, subtraction, multiplication, and division, and are used to simplify and solve problems.
In the given exercises, you see operations in action:

• First, calculating \( f(-3) = 2(-3) + 1 \):
  • Here, multiplication and addition are used to find \(-5\).
• Secondly, substituting into \( g(x) \):
  • \( g(-5) = (-5)^2 - 1 \) uses exponentiation and subtraction to get \(24\).
• Notice how each function involves a set of operations tailored to its form.

Understanding these operations allows you to manipulate and solve functions efficiently. They help transition from a raw mathematical statement to an understandable outcome, providing clarity when composing and evaluating functions.