Problem 57 - Intermediate Algebra

Question

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+x .\) Find each of the following. $$ (f \circ g)(4) $$

Step-by-Step Solution

Answer

The answer is 58.

Step 1: Understand Function Composition

Function composition involves applying one function to the results of another. For two functions, say \(f\) and \(g\), the composition of \(f\) after \(g\) is denoted as \((f \circ g)(x) = f(g(x))\).

Step 2: Apply g(x)

To find \(g(x)\) at \(x = 4\), substitute \(x = 4\) into \(g(x) = x^2 + x\): \[g(4) = 4^2 + 4 = 16 + 4 = 20.\]

Step 3: Substitute g(x) Output into f(x)

Now substitute \(g(4) = 20\) into \(f(x) = 3x - 2\) to find \(f(g(4))\): \[f(g(4)) = f(20) = 3(20) - 2 = 60 - 2 = 58.\]

Key Concepts

Intermediate Algebra

Intermediate algebra is the branch of mathematics that focuses on forming equations with variables, working with functions, and solving different types of algebraic problems. It acts as a stepping stone between basic algebra and more advanced topics like calculus.
To excel in intermediate algebra:

• Understand how to manipulate and solve equations involving variables.
• Be comfortable with operations on polynomials and rational expressions.
• Learn how to graph functions and interpret their meaning.

In the context of the exercise, understanding intermediate algebra is crucial for performing function operations, which includes composition as illustrated with \((f \circ g)(x) = f(g(x))\). This involves substituting one function into another and simplifying the results, a fundamental skill when working with polynomials and their graphs.

Polynomial Functions

Polynomial functions are among the most important expressions in algebra, defined by equations that consist of terms involving variables raised to non-negative integer powers. They are expressed in the form:\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \, \ldots \, + a_2 x^2 + a_1 x + a_0 \]where each \(a_i\) is a constant, and the highest power of \(x\) is the degree of the polynomial.
In polynomial functions:

• The leading coefficient (the coefficient of the term with the highest power) plays a key role in determining the end behavior of its graph.
• They can represent a variety of real-world phenomena, such as projectile motion or population growth.

For the functions \(f(x) = 3x - 2\) and \(g(x) = x^2 + x\) given in the exercise, \(f(x)\) is a linear polynomial (degree 1), and \(g(x)\) is a quadratic polynomial (degree 2). Understanding how to evaluate and manipulate these different types of polynomials is essential in both algebra and its applications.

Function Evaluation

Function evaluation involves finding the output of a function given an input value. This is a primary activity in algebra, especially when working with composite functions. It consists of substituting a specific value for \(x\) in the function's equation and calculating the result.
In our exercise, the process is:

• Evaluate \(g(x)\) at \(x = 4\) by plugging the value into the expression: \(g(x) = x^2 + x\).
• Calculate \(g(4) = 4^2 + 4 = 20\).
• Next, substitute \(g(4) = 20\) into \(f(x) = 3x - 2\) to evaluate the composite function \((f \circ g)(4)\).
• This gives \(f(20) = 3 \times 20 - 2 = 58\).

Evaluating functions effectively requires understanding the function's form, substituting correctly, and following the order of operations. Mastering this skill helps in analyzing and solving complex mathematical problems.