Question

Give the composition of any two functions such that (a) The outside function is a power function and the inside function is a linear function. (b) The outside function is a linear function and the inside function is a power function.

Short answer

Answer: a) The composition of the power function and the linear function is (mx + b)^n, where n is a positive integer and m and b are constants. b) The composition of the linear function and the power function is m(x^n) + b, where n is a positive integer and m and b are constants.

Step-by-step solution

Define the functions

First, let's define a power function and a linear function. A power function can be represented as: f(x) = x^n, where n is a positive integer. A linear function can be represented as: g(x) = mx + b, where m and b are constants.

Compose the functions

Now, we will compose these functions in the following way: (f ∘ g)(x) = f(g(x)). (f ∘ g)(x) = f(mx + b) Now, substitute the power function f(x) = x^n: (f ∘ g)(x) = (mx + b)^n For part (a), the composition of the power function and the linear function is (mx + b)^n, where n is a positive integer and m and b are constants. Now, let's move on to (b). We need to find the composition of a linear function and a power function, such that the linear function is on the outside and the power function is on the inside.

Define the functions (b)

We have already defined a power function and a linear function in the first part. Here they are again for convenience: A power function can be represented as: h(x) = x^n, where n is a positive integer. A linear function can be represented as: k(x) = mx + b, where m and b are constants.

Compose the functions (b)

Now, we will compose these functions in the following way: (k ∘ h)(x) = k(h(x)). (k ∘ h)(x) = k(x^n) Now, substitute the linear function k(x) = mx + b: (k ∘ h)(x) = m(x^n) + b For part (b), the composition of the linear function and the power function is m(x^n) + b, where n is a positive integer and m and b are constants.

Key concepts

Power Function

A power function is an important concept in algebra. It takes the general form of \( f(x) = x^n \), where \( n \) is a positive integer. This function means you raise the input \( x \) to the power of \( n \), which tells us how many times we multiply \( x \) by itself.

Power functions come in various shapes depending on the value of \( n \). For example, when \( n = 2 \), the function is a quadratic function, represented by \( x^2 \), which creates a parabolic graph. Similarly, when \( n = 3 \), it's a cubic function, creating a more complex, but still smooth, curve.

• Quadratic function: \( f(x) = x^2 \)
• Cubic function: \( f(x) = x^3 \)

Understanding how these functions behave helps in graphing and solving equations involving power functions. For compositions, a power function will take an entire expression into a higher-order polynomial when used as the outside function.

Linear Function

Linear functions are the simplest form of algebraic functions, taking the form \( f(x) = mx + b \). Here, \( m \) represents the slope of the line, which is a measure of how steeply the line rises or falls, and \( b \) determines where the line intercepts the y-axis.

Linear functions graph to a straight line and have the property of constant change; for every move along the x-axis, the output changes by a constant amount. For instance, if \( m = 2 \) and \( b = 3 \), every unit increase in \( x \) results in the output increasing by 2.

• Slope \( m \) = rate of change
• Intercept \( b \) = starting value

When used in function composition, linear functions effectively add or scale the complexity of another function, creating either simple linear patterns or modifying more complex functions like power functions.

Algebra and Function Composition

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It allows us to describe mathematical relationships and recognize how different functions can interact.

Function composition is an algebraic concept wherein one function is applied after another. Denoted as \( (f \circ g)(x) = f(g(x)) \), this concept involves using the output of one function \( g \) as the input for another function \( f \). This is akin to putting one process inside another to expand or change the outcomes.

In the context of the exercise, two compositions are demonstrated:

• Power function as outer function: When a power function \( f(x) = x^n \) is applied to a linear function \( g(x) = mx + b \), it results in \( (mx + b)^n \).
• Linear function as outer function: Similarly, when a linear function \( k(x) = mx + b \) is applied to a power function \( h(x) = x^n \), the result is \( m(x^n) + b \).

This technique is powerful in modeling physical phenomena or solving real-world problems where one variable affects another through multiple stages.