Problem 61 - Intermediate Algebra

Question

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+x .\) Find each of the following. $$ (g \circ f)(0) $$

Step-by-Step Solution

Answer

The value of \((g \circ f)(0)\) is 2.

Step 1: Understand the Composition

The symbol \((g \circ f)(x)\) represents the composition of functions \(g\) and \(f\), meaning you first apply function \(f\) to \(x\), and then apply function \(g\) to the result. In this exercise, we need to find \((g \circ f)(0)\).

Step 2: Apply Function f to x=0

First, we need to find \(f(0)\) using the function definition \(f(x) = 3x - 2\). Substitute 0 for \(x\): \[ f(0) = 3(0) - 2 = -2 \]

Step 3: Apply Function g to f(0)

Now that we have \(f(0) = -2\), we apply function \(g\) to this result. Using the function definition \(g(x) = x^2 + x\), substitute -2 for \(x\): \[ g(-2) = (-2)^2 + (-2) = 4 - 2 = 2 \]

Step 4: Combine Results

The composition \((g \circ f)(0)\) is equivalent to \(g(f(0))\), which simplifies to \(g(-2) = 2\). Thus, the value of \((g \circ f)(0)\) is 2.

Key Concepts

Intermediate Algebra Basics

Intermediate algebra is a branch of mathematics focused on manipulating algebraic expressions, equations, and functions. It serves as a bridge between basic algebra and more advanced topics, like calculus. In intermediate algebra, you learn about various functions, how to work with them, and how to evaluate complex expressions. To solve problems involving functions, you often need to:

• Understand how to apply function operations such as addition, subtraction, and composition.
• Manage polynomial expressions and equations efficiently.
• Use substitution to evaluate functions at given points.

Mastering intermediate algebra requires practice with these types of operations, including equations and functions like those given for this exercise.

Understanding Function Evaluation

Function evaluation is an essential skill learned when studying functions in algebra. To evaluate a function, you replace the variable in the function's equation with a specified numerical value. This process helps in understanding how the function behaves for different values of its input.For example, given the function \(f(x) = 3x - 2\), if you want to evaluate \(f\) at \(x = 0\), you substitute 0 in the place of \(x\). This means you calculate:\[f(0) = 3(0) - 2 = -2\]The goal of function evaluation is to determine the output (also known as the functional value) for specific inputs. This gives insight into how the function maps inputs to outputs and is a central element of working with functions in algebra.

The Nature of Composite Functions

A composite function is formed when one function is applied after another. This is represented by the notation \((g \circ f)(x)\), meaning you first apply function \(f\) to the input \(x\), and then apply function \(g\) to the result obtained from \(f\). Composite functions allow you to study how different mathematical operations influence each other.To evaluate a composite function, follow these steps:

• Calculate the inner function's value. For example, find \(f(0)\) if you want to compute \((g \circ f)(0)\).
• Use the output from the inner function as the input for the outer function. For instance, if \(f(0) = -2\), then calculate \(g(-2)\).
• Combine the results to determine the final output of the composition.

In our exercise, the final result of the composition \((g \circ f)(0)\) is \(2\). Understanding composite functions is crucial because they're foundational in mathematics, used in more advanced topics, including calculus and many applied sciences.