Problem 15 - College Algebra

Question

The functions are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$ f(x)=2 x+3 $$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = 2x + 3\) is \(f^{-1}(x) = (x - 3)/2\). The verification process proves that this answer is correct because \(f^{-1}(f(x)) = x\) and \(f(f^{-1}(x)) = x\) for this particular inverse relation.

Step 1: Derive the Inverse Function

To obtain the inverse of a function, one must replace \(f(x)\) with \(y\) and then solve the equation for \(x\). The function of \(f(x)\) is given as \(f(x)=2x+3\), so replacing \(f(x)\) with \(y\) we get \(y = 2x+3\). Rearranging this equation to find \(x\), we will have: \(x = (y-3)/2\). The inverse function \(f^{-1}(x)\) is therefore \(f^{-1}(x)= (x-3)/2\).

Step 2: Verify the first property of the inverse function

Now, insert \(f^{-1}(x)\) into \(f(x)\) to verify the equation \(f(f^{-1}(x))= x\). This means we substitute \(f^{-1}(x)\) into the equation for \(f(x)\) as follows: \(f(f^{-1}(x)) = 2 * (f^{-1}(x)) + 3 = 2 * ((x-3)/2) + 3 =x\). So the first property of the inverse function is verified.

Step 3: Verify the second property of the inverse function

Insert \(f(x)\) into \(f^{-1}(x)\) to verify the equation \(f^{-1}(f(x))=x\). This involves substituting \(f(x)\) into the equation for \(f^{-1}(x)\) as follows: \(f^{-1}(f(x)) = (f(x) - 3)/2 = ((2x+3) - 3)/2 = x\). Hence, the second property of the inverse function is also verified.

Key Concepts

One-to-One Functions

A one-to-one function is a special type of function where each output value corresponds to exactly one input value. This means that no two different inputs can produce the same output. This property is crucial when dealing with inverse functions, because only one-to-one functions can have inverses that are also functions.
Let's consider the given function:

• For example, the function \(f(x) = 2x + 3\) is one-to-one because every different \(x\) gives a different \(f(x)\).

To check if a function is one-to-one, you can use the horizontal line test:

• If no horizontal line cuts the graph of the function more than once, then the function is one-to-one.

In summary, understanding one-to-one functions is vital for comprehending inverse functions as they form the foundational characteristic making the inverse possible.

Verifying Inverse Functions

Once you've found what you assume is the inverse function, it's important to verify it to ensure accuracy. This involves checking two main properties:

• \(f(f^{-1}(x)) = x\)
• \(f^{-1}(f(x)) = x\)

Let's take the function \(f(x) = 2x + 3\) and its inverse \(f^{-1}(x) = \frac{x-3}{2}\):

• For \(f(f^{-1}(x))\), substitute \(f^{-1}(x)\) into \(f(x)\):
• \(f\left(f^{-1}(x)\right) = 2 \left(\frac{x-3}{2}\right) + 3 = x\)

This shows the first property is verified. Now, let's use \(f(x)\) with \(f^{-1}(x)\):

• \(f^{-1}(f(x)) = \frac{(2x + 3) - 3}{2} = x\)

This confirms the second property. Successfully verifying these properties ensures that your inverse is correct.

Algebraic Manipulation

Algebraic manipulation is a fundamental skill when handling inverse functions. It involves rearranging equations to solve for a particular variable. For inverse functions, this process is used to switch the roles of the input and output.
Take the original function \(f(x) = 2x + 3\). To find its inverse, follow these algebraic manipulation steps:

• Replace \(f(x)\) with \(y\): \(y = 2x + 3\).
• Solve for \(x\) by isolating it: \(x = \frac{y - 3}{2}\).
• Swap \(x\) and \(y\) to express the inverse: \(f^{-1}(x) = \frac{x - 3}{2}\).

This step-by-step manipulation is essential in deriving the inverse function. Practice regularly to get comfortable with solving these types of equations.