Question

Find (a) \(f \circ g\) and (b) \(g \circ f\). \(f(x)=\frac{1}{3} x-3, \quad g(x)=3 x+1\)

Short answer

The composition \(f \circ g\) is \(x - \frac{8}{3}\) and the composition \(g \circ f\) is \(x - 8\)

Step-by-step solution

Find \(f \circ g\)

To find the composition \(f \circ g\), the function \(g\) is substituted into the function \(f\). So \(f(g(x))\) will be \(\frac{1}{3}(g(x)) - 3 = \frac{1}{3}(3x+1) - 3\)

Simplify \(f \circ g\)

Now, simplify it to get \(x + \frac{1}{3} - 3 = x - \frac{8}{3}\)

Find \(g \circ f\)

To find the composition \(g \circ f\), the function \(f\) is substituted into the function \(g\). So \(g(f(x))\) will be \(3(f(x)) + 1 = 3(\frac{1}{3}x - 3) + 1\)

Simplify \(g \circ f\)

Now, simplify it to get \(x - 9 + 1 = x - 8\)

Key concepts

Composition of Functions

Understanding the composition of functions is akin to learning how to follow a sequence of instructions one after the other. In algebra, when we talk about composing functions, we mean taking one function and applying it to the results of another function. This operation is denoted by \f \(f \(g(x)\)\) or \f \(f \(g(x)\)\), symbolizing that you first apply \(g\) to \(x\) and then apply \(f\) to the outcome of \(g(x)\).

Imagine you have two machines, where the output of one becomes the input of the other. This chain of processing is precisely what happens during function composition. It is an essential skill in algebra that extends to more complex mathematical concepts and applications.

Algebraic Functions

Algebraic functions serve as the building blocks of algebra. They are expressions using a combination of numbers and at least one variable, connected by operations such as addition, subtraction, multiplication, division, and exponentiation. Functions like \(f(x)=\frac{1}{3} x-3\) and \(g(x)=3 x+1\) in our exercise, are algebraic functions because they map an input \(x\) to a single output based on the given algebraic expression.

An essential characteristic of algebraic functions is their ability to construct rules or mappings from one quantity to another, leading to a range of possibilities in analyzing relationships and finding solutions to problems.

Function Operations

When we discuss function operations in algebra, we're referring to various ways functions can be combined to form new functions. You've already seen an example of function composition, but other operations include addition, subtraction, multiplication, and division of functions. For instance, if we were to add \(f(x)\) and \(g(x)\), we'd simply combine them as \(f(x) + g(x)\). Each operation follows specific rules that dictate how the combined function behaves in relation to its individual parts.

Dealing with these operations requires a methodical approach: substitute accurately, perform the operation, and then simplify the result. Function operations extend the utility of functions, enabling us to model and solve a wider spectrum of mathematical challenges.

Simplifying Expressions

Simplifying expressions is a fundamental technique in algebra that makes complex equations more manageable. It entails reducing equations to their simplest form while maintaining their original value. This process involves combining like terms, using the distributive property, and simplifying fractions.

In our example with function composition, once we substitute one function into another, we simplify the resulting expression to its simplest form, such as reducing \(\frac{1}{3}(3x+1) - 3\) to \(x - \frac{8}{3}\). The goal is to make the expression clearer and easier to work with—whether for graphing, solving for variables, or communicating your solution to others.