Problem 70 - Calculus

Question

Explain what is wrong with the statement. \(f(x)=3 x+5\) and \(g(x)=-3 x-5\) are inverse functions of each other.

Step-by-Step Solution

Answer

The composition of \( f(x) \) and \( g(x) \) does not equal \( x \), so they are not inverses.

Step 1: Define Inverse Functions

Inverse functions are two functions, say \( f(x) \) and \( g(x) \), such that applying \( g \) on the result of \( f \) gives the identity function, i.e., if \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \). This means \( f \) and \( g \) undo each other.

Step 2: Set Up Function Composition

For \( f(x) = 3x + 5 \) and \( g(x) = -3x - 5 \) to be inverses, they must satisfy \( f(g(x)) = x \) and \( g(f(x)) = x \). Let's first explore \( f(g(x)) \).

Step 3: Calculate \( f(g(x)) \)

Substitute \( g(x) = -3x - 5 \) into \( f(x) = 3x + 5 \). We need to calculate \( f(g(x)) = f(-3x - 5) \). \[ f(-3x - 5) = 3(-3x - 5) + 5 = -9x - 15 + 5 = -9x - 10 \] This does not simplify to \( x \).

Step 4: Calculate \( g(f(x)) \)

Now, substitute \( f(x) = 3x + 5 \) into \( g(x) = -3x - 5 \). We need to calculate \( g(f(x)) = g(3x + 5) \).\[ g(3x + 5) = -3(3x + 5) - 5 = -9x - 15 - 5 = -9x - 20 \] This also does not simplify to \( x \).

Step 5: Conclusion on Function Inverses

Since neither \( f(g(x)) = x \) nor \( g(f(x)) = x \), the functions \( f(x) = 3x + 5 \) and \( g(x) = -3x - 5 \) are not inverses of each other.

Key Concepts

Function Composition

When we talk about function composition, we are discussing the process of applying one function to the results of another. Imagine it as feeding the output of one function right into another function as an input. This is denoted as \( f(g(x)) \), meaning we apply \( g \) first and then \( f \).

To check if two functions are inverses, we need to compute both \( f(g(x)) \) and \( g(f(x)) \). If both compositions return the original input \( x \), then the two functions are inverse functions.

• \( f(g(x)) \) means you substitute \( g(x) \) into \( f \).
• \( g(f(x)) \) means you substitute \( f(x) \) into \( g \).

In our exercise, neither \( f(g(x)) \) nor \( g(f(x)) \) simplified back to \( x \), hence \( f(x) \) and \( g(x) \) cannot be inverse functions.

Identity Function

The identity function plays a crucial role in verifying inverse functions. Think of an identity function as a special kind of function that simply returns whatever you input. Mathematically, it's expressed as \( I(x) = x \).

For two functions to be inverses, the result of their composition must be the identity function. Specifically, when you compose \( f(g(x)) \) or \( g(f(x)) \), the output should always be \( x \).

• The identity function is essentially the gold standard for inverse function verification.
• It ensures the operation "undoes" the effect of the first function.

In our specific scenario, because neither \( f(g(x)) \) nor \( g(f(x)) \) equaled \( x \), the functions are not inverses, demonstrating there is an issue with identifying them as such.

Linear Functions

Linear functions are perhaps the simplest form of functions you will encounter. They are typically written as \( f(x) = ax + b \), where \( a \) and \( b \) are constants. This form represents a straight line when graphed.

To identify whether two linear functions are inverses of each other, it's essential to check both the slopes and the y-intercepts. Actually, two linear functions can potentially be inverses if:

• The coefficients \( a \) and \( -a \) are multiplicative inverses of each other.
• When composed, their y-intercepts effectively cancel out and simplify to \( 0 \).

In our exercise, the linear functions \( f(x) = 3x + 5 \) and \( g(x) = -3x - 5 \) do not satisfy these conditions, as demonstrated by the calculations showing neither composite function simplifies to \( x \). Therefore, they are not inverses.