Question

Give three different composite functions with the property that the outside function raises the inside function to the third power.

Short answer

Answer: Three different composite functions that satisfy this condition are: 1. \((g \circ f)(x) = (x + 1)^3\) 2. \((g \circ f)(x) = -x^3\) 3. \((g \circ f)(x) = (2x - 1)^3\)

Step-by-step solution

First Composite Function

Let's create our first composite function. We can start by defining the inside function as \(f(x) = x + 1\). Then, we define the outside function as \(g(x) = x^3\). Now, we can create the composite function \((g \circ f)(x)\): \((g \circ f)(x) = g(f(x)) = g(x + 1) = (x + 1)^3\).

Second Composite Function

Now, let's create our second composite function. Define the inside function as \(f(x) = -x\). Then, define the outside function as \(g(x) = x^3\). Again, we can create the composite function \((g \circ f)(x)\): \((g \circ f)(x) = g(f(x)) = g(-x) = (-x)^3 = -x^3\).

Third Composite Function

Finally, let's create our third composite function. Define the inside function as \(f(x) = 2x - 1\). Define the outside function as \(g(x) = x^3\). Now, we can create the composite function \((g \circ f)(x)\): \((g \circ f)(x) = g(f(x)) = g(2x - 1) = (2x - 1)^3\). So, we have found three different composite functions with the property that the outside function raises the inside function to the third power: 1. \((g \circ f)(x) = (x + 1)^3\) 2. \((g \circ f)(x) = -x^3\) 3. \((g \circ f)(x) = (2x - 1)^3\)

Key concepts

Outside Function

When it comes to composite functions, the outside function is what you apply last. Think of it as the final touch in a sequence of operations. In the task of creating composite functions, the outside function consistently used is the exponentiation to the power of three. Here, the role of the outside function is crucial because it determines the characteristic of the output based on the input processed by the inside function. For the problem at hand, the outside function, denoted usually by \( g(x) \), is always \( x^3 \). When you see anything raised to the third power, it means this function takes a number and multiplies it by itself three times. This constant role of \( g(x) = x^3 \) ensures that any alterations made by different inside functions are ultimately raised to the third power, thereby creating the desired composite function.

Inside Function

The inside function is what you apply first when dealing with composite functions. It's the part where the initial action takes place on the input variable before handing it over to the outside function. In our problem, different inside functions alter the input in unique ways before it's handed over to the outside exponentiation process. Here's a breakdown:

• For the first composite function, the inside function is \( f(x) = x + 1 \). This shifts the input by adding one.
• In the second composite function, \( f(x) = -x \) multiplies the input by -1, flipping its sign.
• The third function uses \( f(x) = 2x - 1 \), which scales the input by 2 and then subtracts 1.

Each inside function modifies \( x \) in a specific way, which preps it before it's placed into the outside function for the final operation. Understanding how these inside functions work is key to predicting how the composition will behave.

Function Composition

Function composition is like building a pipeline where outputs from one function become inputs to another. When composing two functions, you are essentially chaining them together into a new operation that reflects the sequential processing of data.In mathematical notation, if we have functions \( f(x) \) and \( g(x) \), their composition is expressed as \((g \circ f)(x)\), which means \( g(f(x)) \). Here's a step-by-step process of what happens during function composition:

• Begin with an initial input \( x \).
• Apply the inside function \( f \) to \( x \), transforming the input.
• The result, \( f(x) \), is then passed to the outside function \( g \).
• The final output is \( g(f(x)) \), the value of the entire composite function.

This process allows us to combine simple functions into more complex ones, enabling us to perform intricate calculations through a series of simpler steps. By understanding each function's role and effect, we harness the power of composition to create versatile mathematical tools.