Problem 24 - Algebra A Combined Function

Question

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=7 x-1 ; g(x)=\sqrt[3]{x} $$

Step-by-Step Solution

Answer

(f ∘ g)(x) = 7x^(1/3) - 1; (g ∘ f)(x) = ∛(7x - 1).

Step 1: Understanding Function Composition

Function composition \( (f \circ g)(x) \) means applying function \( g(x) \) first, and then applying function \( f(x) \) to the result of \( g(x) \). In this exercise, we'll substitute the expression for \( g(x) \) into \( f(x) \). Similarly, \( (g \circ f)(x) \) involves substituting \( f(x) \) into \( g(x) \).

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Step 6: Conclude with the answer

(f ∘ g)(x) = 7x^(1/3) - 1; (g ∘ f)(x) = ∛(7x - 1).

Key Concepts

algebra

Algebra is like the language of mathematics. It helps us understand and solve problems by using symbols and letters to represent numbers. When working with algebra, you can think of it as a kind of puzzle
—you take apart each piece to understand how everything fits together.
In algebra:

• We use variables for unknown values, like \( x \) or \( y \).
• We perform operations such as addition, subtraction, multiplication, and division.
• We solve equations by finding the value of the variables that make the equation true.

Algebra helps us see relationships between numbers and find the solutions to complex problems. It's a fundamental building block for more advanced topics in math, like mathematical functions.

mathematical functions

Mathematical functions are like machines that take an input, do something to it, and give an output. Think of each function as having a special rule that tells it what to do with the input. When you see a function, you're looking at a process that converts an input, often a number, into an output using a specific formula.
For example:

• In the function \( f(x) = 7x - 1 \), the rule is to multiply the input by 7 and then subtract 1 from the result.
• In the function \( g(x) = \sqrt[3]{x} \), the rule is to find the cube root of the input \( x \).

Functions are essential in algebra because they allow us to model and analyze the relationships between different quantities. When you combine or compose functions, you create more complex rules by stacking these smaller rules on top of each other.

composing functions

Composing functions is the process of combining two functions to create a new function. It's like putting together different machines to perform a more complex task. In algebra, when we compose functions, we feed the output of one function into another.
Here's how it works with our functions \( f(x) = 7x - 1 \) and \( g(x) = \sqrt[3]{x} \):

• To find \( (f \circ g)(x) \), substitute \( g(x) \) into \( f(x) \). That means wherever there's an \( x \) in \( f(x) \), replace it with \( g(x) = \sqrt[3]{x} \), so \( (f \circ g)(x) = 7(\sqrt[3]{x}) - 1 \).
• To find \( (g \circ f)(x) \), swap the process. Substitute \( f(x) \) into \( g(x) \). That means replace \( x \) in \( g(x) \) with \( f(x) = 7x - 1 \), so \( (g \circ f)(x) = \sqrt[3]{7x - 1} \).

Composing functions is a powerful tool because it lets us build complex expressions that can model real-world scenarios. It shows how different processes and rules work together to transform inputs into outputs in a structured way.