Question

If \(f(x)=5 x^{2}+4 x-8\) and \(g(x)=3 x-1,\) find \((f \circ g)(x)\)

Short answer

(f ∘ g)(x) = 45x^2 - 18x - 7.

Step-by-step solution

Understand the Composition of Functions

To find \( (f \circ g)(x) \), we need to substitute \( g(x) \) into \( f(x) \). This means we will evaluate \( f(g(x)) \).

Identify \( g(x) \)

The function \( g(x) \) is given as \( g(x) = 3x - 1 \).

Substitute \( g(x) \) into \( f(x) \)

Substitute \( g(x) = 3x - 1 \) into \( f(x) \). This means we will replace each \( x \) in \( f(x) \) with \( 3x - 1 \). So, \( f(g(x)) = f(3x - 1) \).

Apply the Substitution in \( f(x) \)

The function \( f(x) \) is given as \( f(x) = 5x^2 + 4x - 8 \). Substitute \( x = 3x - 1 \), so \( f(3x - 1) = 5(3x - 1)^2 + 4(3x - 1) - 8 \).

Calculate \( (3x - 1)^2 \)

Calculate the square: \( (3x - 1)^2 = 9x^2 - 6x + 1 \).

Substitute Back into \( f(x) \)

Using the result from the previous step, substitute: \( f(3x - 1) = 5(9x^2 - 6x + 1) + 4(3x - 1) - 8 \).

Simplify the Expression

Expand and simplify: \( 5(9x^2 - 6x + 1) + 4(3x - 1) - 8 = 45x^2 - 30x + 5 + 12x - 4 - 8 \).

Combine Like Terms

Combine like terms to simplify further: \( 45x^2 -18x - 7 \).

Key concepts

Function Composition

To understand function composition, let's break it down simply. Function composition is about combining two functions where the output of one function becomes the input of another. In mathematics, this is denoted as \( (f \circ g)(x) \). Here, it means we first apply function \( g(x) \) and then apply function \( f(x) \) to the result. Think of it like a two-step process. It’s important to follow the correct order: first substitute \( g(x) \) into any \( x \) that appears in \( f(x) \). This ensures the problems are solved in a proper sequential manner.

Substitution

Substitution is a fundamental method in algebra used for solving functions. It involves replacing a variable with a given expression. For instance, if we know \( g(x) = 3x - 1 \) and need to find \( f(g(x)) \), we replace every instance of \( x \) in \( f(x) \) with \( 3x - 1 \). So, if \( f(x) = 5x^2 + 4x - 8 \), substitution will transform it into \( f(3x - 1) = 5(3x - 1)^2 + 4(3x - 1) - 8 \). This step-by-step replacement is crucial for solving composite functions correctly.

Quadratic Functions

Quadratic functions have the general form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In our example, \( f(x) = 5x^2 + 4x - 8 \) is a quadratic function. When dealing with function composition involving quadratic functions, you’ll need to handle polynomials. After substitution, the new expression often requires expansion and simplification. For example, \( (3x - 1)^2 \) expands to \( 9x^2 - 6x + 1 \). Understanding how to manipulate these terms correctly is essential.

Simplifying Expressions

Simplifying expressions is the final and often crucial step in solving composite functions. This involves combining like terms and simplifying the polynomial to its simplest form. In the given solution, once we substitute and expand, we reach \( f(3x - 1) = 5(9x^2 - 6x + 1) + 4(3x - 1) - 8 \). Expanding and simplifying gives us \( 45x^2 - 30x + 5 + 12x - 4 - 8 \), which simplifies further to \( 45x^2 - 18x - 7 \). Always combine like terms and watch for arithmetic mistakes to ensure your final expression is correct.