Problem 80 - Essential Calculus Early Transcendentals

Question

(a) If we shift a curve to the left, what happens to its reflection about the line \(y=x ?\) In view of this geometric principle, find an expression for the inverse of \(g(x)=f(x+c),\) where \(f\) is a one-to-one function. (b) Find an expression for the inverse of \(h(x)=f(c x)\) , where \(c \neq 0\)

Step-by-Step Solution

Answer

(a) The inverse of \(g(x)=f(x+c)\) is \(g^{-1}(x) = f^{-1}(x) - c\). (b) The inverse of \(h(x)=f(cx)\) is \(h^{-1}(x) = f^{-1}\left(\frac{x}{c}\right)\)."

Step 1: Understanding the Problem: Part (a)

We want to understand the effect of horizontally shifting a function on its inverse. Given a function shift by changing it to \(f(x+c)\), we need to see what happens when we reflect this shifted curve about the line \(y=x\) to find the inverse.

Step 2: Analyzing Part (a)

The reflection of a curve about the line \(y = x\) is the inverse of the function. Shifting a function \(f(x)\) horizontally to the left by \(c\) units gives \(g(x) = f(x+c)\), so we need to find an inverse of this function.

Step 3: Finding the Inverse for Part (a)

The inverse of a function \(y = f(x+c)\) can be derived by replacing \(y\) with \(x\), so if any point \((a, b)\) is on the curve of \(f(x+c)\), then \((b, a)\) is on the inverse curve. Thus, \(x = f(y+c)\), and solving for \(y\) gives \(y = f^{-1}(x) - c\). Therefore, the inverse is \(g^{-1}(x) = f^{-1}(x) - c\).

Step 4: Understanding the Problem: Part (b)

Now, we'll address scaling by a constant factor within the function. For \(h(x)=f(cx)\), we are asked to find its inverse function.

Step 5: Finding the Inverse for Part (b)

The inverse of \(y = f(cx)\) requires swapping \(x\) and \(y\) to find \(cx = f^{-1}(y)\). Solving for \(y\), we get \(y = f^{-1}\left(\frac{x}{c}\right)\). Thus, the inverse of \(h(x)\) is \(h^{-1}(x) = f^{-1}\left(\frac{x}{c}\right)\).

Key Concepts

Function Reflection

Understanding function reflection can be super helpful when dealing with inverse functions. When we reflect a function over a specific line, in this case, the line \(y = x\), we essentially perform a special type of transformation. Let's break it down.

For any function \(f(x)\), reflecting it over the line \(y = x\) means swapping each \(x\)-value with its corresponding \(y\)-value. This simplifies to switching the roles of the input and output variables.

By doing this swap, you create what's known as the inverse of the function, denoted by \(f^{-1}(x)\). This means if you started with a point \((a, b)\) on the graph of \(f(x)\), after reflection, that point turns into \((b, a)\).

• Reflection transforms: swap \(x\) and \(y\)
• Creating an inverse function
• Helps in finding the reversed operations of the function

By using reflection, you're essentially unraveling the function to see how it behaves in reverse. It’s a powerful concept, especially useful for unraveling and understanding inverse functions.

Horizontal Shift

A horizontal shift involves moving the whole graph of a function to the left or right along the x-axis. Imagine a simple sliding motion, much like pushing a toy car back and forth - it's straightforward.

When you shift a function like \(f(x)\) by adding or subtracting a constant from \(x\), you get a new function, \(f(x+c)\) if shifted to the left, or \(f(x-c)\) if shifted to the right.

For example:

• Shifting left: Replace \(x\) with \(x+c\) (negative shift)
• Shifting right: Replace \(x\) with \(x-c\) (positive shift)

How does this affect the structure of the inverse? When you horizontally shift \(f(x)\) left, to find its inverse, you find \(g^{-1}(x) = f^{-1}(x) - c\). This means you essentially need to shift back when finding the inverse function.
You undo the horizontal shift by changing the sign again. So simple but crucial in keeping track of function transformations!

Inverse Calculation

Calculating an inverse function is like doing everything backward. To find the inverse of a function, you're essentially trying to find a function that "undoes" what the original function does.

Here's a step-by-step guide:

• Swap the variables: Replace \(y\) with \(x\) to transform the equation, reflecting the function.
• Solve for the new \(y\): Rearrange the equation to solve for \(y\) in terms of \(x\).
• Check the work: Ensure the steps make sense and match what an inverse should do, changing the role of inputs and outputs.

Consider the function \(g(x) = f(x+c)\).
When finding its inverse, you adjust for the horizontal shift to restore its original position. This means calculating \(g^{-1}(x) = f^{-1}(x) - c\). This step corrects for the shifted starting point.

For another example, with \(h(x) = f(cx)\), start by replacing \(x\) with \(y\) and solve for \(y\): you end up with \(h^{-1}(x) = f^{-1}\left(\frac{x}{c}\right)\). This solves the mystery of reversing function actions; ensuring you correctly find its inverse.