Question

Use the functions \(f(x)=3 x, g(x)=x-7\), and \(h(x)=x^{2}\) to determine each of the following. a) \((f \circ g \circ h)(x)\). b) \(g(f(h(x)))\) c) \(f(h(g(x)))\) d) \((h \circ g \circ f)(x)\)

Short answer

a) 3x^2 - 21, b) 3x^2 - 7, c) 3x^2 - 14x + 147, d) 9x^2 - 42x + 49.

Step-by-step solution

Understand the Functions

We have three functions: \( f(x) = 3x \), \( g(x) = x - 7 \), and \( h(x) = x^2 \). We need to find compositions of these functions in different orders.

Find \((f \, \circ \, g \, \circ \, h)(x)\)

First, find \((g \, \circ \, h)(x)\). \[ g(h(x)) = g(x^2) = x^2 - 7 \] Now find \((f \, \circ \, g \, \circ \, h)(x)\). \[ f(g(h(x))) = f(x^2 - 7) = 3(x^2 - 7) = 3x^2 - 21 \]

Find \(g(f(h(x)))\)

First, find \(h(x)\). \[ h(x) = x^2 \] Next, apply \(f\) to \(h(x)\). \[ f(h(x)) = f(x^2) = 3x^2 \] Now apply \(g\) to \(f(h(x))\). \[ g(f(h(x))) = g(3x^2) = 3x^2 - 7 \]

Find \(f(h(g(x)))\)

First, find \(g(x)\). \[ g(x) = x - 7 \] Next, apply \(h\) to \(g(x)\). \[ h(g(x)) = h(x - 7) = (x - 7)^2 = x^2 - 14x + 49 \] Now apply \(f\) to \(h(g(x))\). \[ f(h(g(x))) = f(x^2 - 14x + 49) = 3(x^2 - 14x + 49) = 3x^2 - 42x + 147 \]

Find \((h \, \circ \, g \, \circ \, f)(x)\)

First, find \(f(x)\). \[ f(x) = 3x \] Next, apply \(g\) to \(f(x)\). \[ g(f(x)) = g(3x) = 3x - 7 \] Now apply \(h\) to \(g(f(x))\). \[ h(g(f(x))) = h(3x - 7) = (3x - 7)^2 = 9x^2 - 42x + 49 \]

Key concepts

precalculus

Precalculus is a foundational course that prepares students for the study of calculus. It covers various topics, including functions, sequences, series, and trigonometry.
Understanding functions and their compositions is essential in precalculus, as it lays the groundwork for more advanced mathematical concepts.
In this exercise, we explore function configurations, which is a common precalculus topic used to combine different functions into a single, new function.

composite functions

Composite functions involve combining two or more functions to create a single function. This process is called function composition and is represented using the symbol '∘'.
For example, if we have two functions, \(f(x)\) and \(g(x)\), their composition is written as \((f ∘ g)(x)\). This means we first apply \(g(x)\) and then apply \(f(x)\) to the result.
In the given exercise, we practiced finding compositions of three functions: \(f(x)\), \(g(x)\), and \(h(x)\). This involves steps like \((f ∘ g ∘ h)(x)\), \(g(f(h(x)))\), and others.
Mastering composite functions is crucial for understanding more complex scenarios in calculus.

function operations

Function operations involve combining functions using various arithmetic operations such as addition, subtraction, multiplication, and division.
In the exercise, we deal specifically with composition operations. When performing composite functions, follow these steps:

• Identify the innermost function first.
• Evaluate the innermost function.
• Use the result as the input for the next function in the composition.
• Repeat the process until the composition is complete.

For instance:
To find \((f ∘ g ∘ h)(x)\) for \(f(x)=3x\), \(g(x)=x-7\), and \(h(x)=x^2\), follow:
1. Find \(h(x) = x^2\)
2. Apply \(g\) to the result: \(g(h(x)) = g(x^2) = x^2 - 7\)
3. Finally, apply \(f\): \(f(g(h(x))) = f(x^2 - 7) = 3(x^2 - 7) = 3x^2 - 21\).

mathematical notation

Mathematical notation provides a systematic way to represent mathematical concepts. It's essential for clear communication and understanding in mathematics.
In the context of the exercise, we use notation for functions and composites:

• \(f(x)\) denotes a function named \(f\) with the variable \(x\).
• \((f ∘ g)(x)\) denotes the composition of \(f(x)\) and \(g(x)\), meaning \(f(g(x))\).
• Use of the parentheses to clearly indicate function inputs and order of operations.

Let's see some examples:
For \(f(x) = 3x\), \(g(x) = x - 7\), and \(h(x) = x^2\):
1. \(g(h(x)) = g(x^2) = x^2 - 7\)
2. \(f(g(h(x))) = f(x^2 - 7) = 3(x^2 - 7) = 3x^2 - 21\)
Proper mathematical notation ensures that these operations are easy to follow and understand.