Problem 40 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

(f \circ g)(-3) = 17.

Step 1: Understand Function Composition

Function composition involves applying one function to the results of another. In notation, \( (f \circ g)(x) \) means you first apply \( g(x) \) and then apply \( f(x) \) to the result of \( g(x) \). It can be written as \( f(g(x)) \).

Step 2: Evaluate g(x) at x = -3

Start by substituting \( x = -3 \) into the function \( g(x) = x^2 - 1 \).Substitute the value: \[ g(-3) = (-3)^2 - 1 = 9 - 1 = 8 \]

Step 3: Evaluate f(g(x)) Using the Result from Step 2

Now use the result from Step 2 as the input for \( f(x) \). So, compute \( f(8) \).Substitute the value into the function \( f(x) = 2x + 1 \): \[ f(8) = 2(8) + 1 = 16 + 1 = 17 \]

Step 4: Write Down the Result

The value of the composition \( (f \circ g)(-3) \) is the result computed in the previous step. Therefore, \( (f \circ g)(-3) = 17 \).

Key Concepts

Evaluation of Functions

When working with functions, it's important to understand what it means to evaluate them. Function evaluation simply involves calculating the output of a function given a specific input. To evaluate a function, follow these steps:

• Identify the function's formula or expression, such as \( f(x) \) or \( g(x) \).
• Substitute the given input value into the function instead of the variable. Replace the variable \( x \) with the specific number given.
• Perform the operations indicated by the function's expression to discover the output.

Evaluating a function provides a specific output that can be used in various mathematical applications, such as graphing equations or composing multiple functions. These steps are fundamental when you need to evaluate results within complex operations like function composition.

Substitution

Substitution is a mathematical operation where you replace a variable with its actual value to simplify expressions or find values. This technique is crucial in many mathematical contexts, including algebra and calculus. In the context of evaluating or composing functions, substitution involves:

• Replacing the variable in the function's expression with a specific numerical value provided in the problem.
• Simplifying the expression after the substitution to reach a final numerical result.

Substitution in function composition, like the exercise with \((f \circ g)(-3)\), involves first evaluating the inner function \( g(x) \) by substituting \( x \) with \(-3\). Then, the result from \( g(x) \) is used as the input in \( f(x) \), where substitution occurs again. This step-by-step substitution ensures that each function is calculated in order and correctly applied.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. These are the building blocks for functions and are used when creating equations or during problem-solving in algebra.When you encounter an algebraic expression like \( f(x) = 2x + 1 \), it involves:

• Numbers, which are constant values such as 1 or 2.
• Variables, often represented by letters like \( x \), which stand in for unknown numbers or values that can change.
• Operations, such as addition, subtraction, multiplication, or division, that are applied to the numbers and variables.

Working with algebraic expressions involves understanding the order of operations and how to manipulate them through operations like substitution and evaluation. Grasping these concepts is essential for successfully solving mathematical procedures, as seen in composing functions like \((f \circ g)(x)\), where algebraic manipulation starts with correctly performing the operations within these expressions.