Problem 67 - Algebra 2

Question

For each pair of functions, find \(f(g(x))\) and \(g(f(x))\) $$ f(x)=-x-7, g(x)=4 x $$

Step-by-Step Solution

Answer

Therefore, the composition of functions \(f\) and \(g\) gives \(f(g(x)) = -4x - 7\) and \(g(f(x)) = -4x - 28.\)

Step 1: Finding \(f(g(x))\)

To find \(f(g(x))\), substitute \(g(x)\) into \(f(x)\). This gives us the expression \(f(g(x)) =-(4x)-7= -4x-7.\)

Step 2: Finding \(g(f(x))\)

Similarly, to find \(g(f(x))\), substitute \(f(x)\) into \(g(x)\). This gives us the expression \(g(f(x)) =4(-x-7)= -4x-28.\)

Key Concepts

Function Composition

Function composition is when one function is applied to the result of another function. It is like having two processes, and using the output of the first process as the input for the second. This is expressed as \((f \circ g)(x)\), which means "apply \(g\) first, then apply \(f\)".

In this exercise, we have two functions, \(f(x)=-x-7\) and \(g(x)=4x\). To find \(f(g(x))\):

• First, find the output of \(g(x)\), which is \(4x\).
• Then, substitute \(4x\) into \(f(x)\), replacing every \(x\) in \(f(x)\) with \(4x\).
• This gives us \(f(g(x)) = -(4x) - 7 = -4x - 7\).

Function composition helps to streamline complex processes and create a new function from existing ones. It is useful in mathematics, programming, and various scientific fields.

Inverse Functions

Inverse functions "undo" each other. If you have a function \(f(x)\), its inverse \(f^{-1}(x)\) reverses the process. Think of it as working backwards from the output to the input.

Finding an inverse function involves switching the roles of \(x\) and \(y\). You solve the equation \(y = f(x)\) for \(x\). While function composition takes you one way, an inverse function takes you back.

For example, if \(f(x) = y\) and the inverse \(f^{-1}(y) = x\), then applying \(f\) and then \(f^{-1}\) will get you back to your original starting value. In the exercise case, knowing how to find an inverse helps understand why compositions can be simplified or "undone" in certain situations.

Domain and Range

Understanding the domain and range of functions is essential for function composition. The domain of a function is the set of all possible inputs, while the range is the set of all possible outputs.

When composing functions, it is crucial to consider:

• The domain of the first function (\(g(x)\) in \(f(g(x))\)).
• Then, check if the range of \(g(x)\) falls within the domain of \(f(x)\).

This ensures that each input into the composed function provides a valid output.

In our example, \(g(x) = 4x\) has a domain and range of all real numbers, which fits the domain requirements of \(f(x) = -x - 7\). Therefore, the function composition \(f(g(x))\) is valid with these given functions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. Understanding how to manipulate these expressions is crucial for composing functions.

In this exercise, to find \(g(f(x))\), you compose \(g\) and \(f\):

• Take the expression for \(f(x) = -x - 7\).
• Substitute it into \(g(x) = 4x\), so that each \(x\) in \(g(x)\) is replaced by \(-x - 7\).
• This results in the expression \(g(f(x)) = 4(-x - 7)\), which simplifies to \(-4x - 28\).

Mastering algebraic expressions allows you to effectively perform such substitutions, simplifying the composition and manipulation of functions.