For Which Pair Of Functions Is \[$(g \circ F)(a) = |a| - 2\$\]?A. \[$f(a) = A^2 - 4\$\] And \[$g(a) = \sqrt{a}\$\]B. \[$f(a) = \frac{1}{2} A - 1\$\] And \[$g(a) = 2a - 2\$\]C. \[$f(a) = 5 + A^2\$\] And

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Introduction

In mathematics, the composition of functions is a fundamental concept that plays a crucial role in various areas of study, including algebra, calculus, and analysis. The composition of two functions, denoted as {(g \circ f)(a)$}$, is defined as the function obtained by applying the first function, {f(a)$}$, and then applying the second function, {g(a)$}$, to the result. In this article, we will explore the problem of finding the pair of functions for which the composition {(g \circ f)(a) = |a| - 2$}$ holds true.

Understanding the Composition of Functions

Before we dive into the problem, let's take a moment to understand the composition of functions. Given two functions, {f(a)$}$ and {g(a)$}$, the composition {(g \circ f)(a)$}$ is defined as:

{(g \circ f)(a) = g(f(a))$}$

This means that we first apply the function {f(a)$}$ to the input {a$}$, and then apply the function {g(a)$}$ to the result.

Analyzing the Given Equation

The given equation is {(g \circ f)(a) = |a| - 2$}$. To find the pair of functions that satisfies this equation, we need to analyze the properties of the absolute value function, {|a|$}$. The absolute value function is defined as:

{|a| = \begin{cases} a & \text{if } a \geq 0 \ -a & \text{if } a < 0 \end{cases}$}$

Option A: {f(a) = a^2 - 4$}$ and {g(a) = \sqrt{a}$}$

Let's start by analyzing the first option:

{f(a) = a^2 - 4$}$

{g(a) = \sqrt{a}$}$

To find the composition {(g \circ f)(a)$}$, we need to apply the function {f(a)$}$ first and then apply the function {g(a)$}$ to the result.

{(g \circ f)(a) = g(f(a)) = g(a^2 - 4) = \sqrt{a^2 - 4}$}$

However, this expression does not match the given equation {(g \circ f)(a) = |a| - 2$}$. Therefore, option A is not the correct pair of functions.

Option B: {f(a) = \frac{1}{2} a - 1$}$ and {g(a) = 2a - 2$}$

Let's analyze the second option:

{f(a) = \frac{1}{2} a - 1$}$

{g(a) = 2a - 2$}$

To find the composition {(g \circ f)(a)$}$, we need to apply the function {f(a)$}$ first and then apply the function {g(a)$}$ to the result.

{(g \circ f)(a) = g(f(a)) = g(\frac{1}{2} a - 1) = 2(\frac{1}{2} a - 1) - 2 = a - 4$}$

However, this expression does not match the given equation {(g \circ f)(a) = |a| - 2$}$. Therefore, option B is not the correct pair of functions.

Option C: {f(a) = 5 + a^2$}$ and {g(a) = \sqrt{a}$}$

Let's analyze the third option:

{f(a) = 5 + a^2$}$

{g(a) = \sqrt{a}$}$

To find the composition {(g \circ f)(a)$}$, we need to apply the function {f(a)$}$ first and then apply the function {g(a)$}$ to the result.

{(g \circ f)(a) = g(f(a)) = g(5 + a^2) = \sqrt{5 + a^2}$}$

However, this expression does not match the given equation {(g \circ f)(a) = |a| - 2$}$. Therefore, option C is not the correct pair of functions.

Conclusion

After analyzing all three options, we can conclude that none of them satisfy the given equation {(g \circ f)(a) = |a| - 2$}$. However, we can try to find a different pair of functions that satisfies the equation.

Alternative Solution

Let's consider a different pair of functions:

{f(a) = a - 2$}$

{g(a) = |a| + 2$}$

To find the composition {(g \circ f)(a)$}$, we need to apply the function {f(a)$}$ first and then apply the function {g(a)$}$ to the result.

{(g \circ f)(a) = g(f(a)) = g(a - 2) = |a - 2| + 2$}$

Using the definition of the absolute value function, we can rewrite this expression as:

{|a - 2| + 2 = \begin{cases} a - 2 + 2 & \text{if } a - 2 \geq 0 \ -(a - 2) + 2 & \text{if } a - 2 < 0 \end{cases}$}$

Simplifying this expression, we get:

{|a - 2| + 2 = \begin{cases} a & \text{if } a \geq 2 \ -a + 4 & \text{if } a < 2 \end{cases}$}$

This expression matches the given equation {(g \circ f)(a) = |a| - 2$}$. Therefore, the pair of functions {f(a) = a - 2$}$ and {g(a) = |a| + 2$}$ satisfies the equation.

Final Answer

The final answer is: D\boxed{D}

Introduction

In our previous article, we explored the problem of finding the pair of functions for which the composition {(g \circ f)(a) = |a| - 2$}$ holds true. We analyzed three options and found that none of them satisfied the equation. However, we were able to find an alternative solution by considering a different pair of functions.

In this article, we will answer some frequently asked questions related to the composition of functions and absolute value.

Q: What is the composition of functions?

A: The composition of two functions, denoted as {(g \circ f)(a)$}$, is defined as the function obtained by applying the first function, {f(a)$}$, and then applying the second function, {g(a)$}$, to the result.

Q: How do you find the composition of two functions?

A: To find the composition of two functions, you need to apply the first function, {f(a)$}$, first and then apply the second function, {g(a)$}$, to the result.

Q: What is the absolute value function?

A: The absolute value function, denoted as {|a|$}$, is defined as:

{|a| = \begin{cases} a & \text{if } a \geq 0 \ -a & \text{if } a < 0 \end{cases}$}$

Q: How do you simplify the composition of functions involving absolute value?

A: To simplify the composition of functions involving absolute value, you need to consider the definition of the absolute value function and apply it to the result.

Q: Can you provide an example of a pair of functions that satisfies the equation {(g \circ f)(a) = |a| - 2$}$?

A: Yes, one example of a pair of functions that satisfies the equation is:

{f(a) = a - 2$}$

{g(a) = |a| + 2$}$

Q: How do you determine if a pair of functions satisfies the equation {(g \circ f)(a) = |a| - 2$}$?

A: To determine if a pair of functions satisfies the equation, you need to apply the first function, {f(a)$}$, and then apply the second function, {g(a)$}$, to the result. If the result matches the given equation, then the pair of functions satisfies the equation.

Q: Can you provide a step-by-step solution to the problem?

A: Yes, here is a step-by-step solution to the problem:

  1. Define the pair of functions {f(a)$}$ and {g(a)$}$.
  2. Apply the first function, {f(a)$}$, to the input {a$}$.
  3. Apply the second function, {g(a)$}$, to the result.
  4. Simplify the expression using the definition of the absolute value function.
  5. Check if the result matches the given equation {(g \circ f)(a) = |a| - 2$}$.

Q: What are some common mistakes to avoid when working with the composition of functions and absolute value?

A: Some common mistakes to avoid when working with the composition of functions and absolute value include:

  • Not considering the definition of the absolute value function.
  • Not applying the functions in the correct order.
  • Not simplifying the expression correctly.
  • Not checking if the result matches the given equation.

Conclusion

In this article, we answered some frequently asked questions related to the composition of functions and absolute value. We provided examples and step-by-step solutions to help readers understand the concepts better. We also highlighted some common mistakes to avoid when working with the composition of functions and absolute value.