Problem 23 - 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 $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$ f(x)=\sqrt{x} $$

Step-by-Step Solution

Answer

\(f^{-1}(x)=x^{2}\)

Step 1: Finding the Inverse Function (f^{-1}(x))

To find the inverse of a function \(f(x)\), take 'y=\(f(x)\)' to be the function in question and solve for 'x'. So starting with \(y=\sqrt{x}\), switch 'x' and 'y' to get \(x=\sqrt{y}\), and then square both sides to get rid of the square root, providing the equation for \(f^{-1}(x)\) as \(f^{-1}(x)=x^{2}\).

Step 2: Verify f(f^{-1}(x))=x

Substitute \(f^{-1}(x)\) into \(f(x)\) with the equation \(f(f^{-1}(x))=x. This means replacing every occurrence of 'x' in the function \(f(x)\) with the expression for \(f^{-1}(x)\), which after solving gives us \(f(f^{-1}(x))=\sqrt{x^{2}}=x\).

Step 3: Verify f^{-1}(f(x))=x

Substitute \(f(x)\) into \(f^{-1}(x)\) with the equation \(f^{-1}(f(x))=x. This means replacing every occurrence of 'x' in \(f^{-1}(x)\) with the expression for \(f(x)\). That gives us \(f^{-1}(f(x))=(\sqrt{x})^{2}=x\).

Key Concepts

One-to-one Functions

In mathematics, a function is called one-to-one (also known as injective) if every output value is paired with exactly one input value. This means no two different input values map to the same output value. This property is crucial when trying to find an inverse function, as only one-to-one functions can have inverses that are also functions.

To determine if a function is one-to-one, you can use the horizontal line test: if any horizontal line crosses the graph of the function at most once, the function is one-to-one.

• If a function is one-to-one, it passes this test and thus has an inverse.
• If a function isn't one-to-one, it fails this test and cannot have an inverse that is also a function.

Understanding the nature of one-to-one functions helps in easily identifying which functions can have inverses, laying the groundwork for more complex function operations.

Verification of Inverse Functions

Verifying inverse functions involves checking that a function and its inverse, when composed, yield the identity function. This means that when one function is applied to the result of the other, they essentially "undo" each other, leaving you with the original value.

To verify that a function \( f(x) \) and its inverse \( f^{-1}(x) \) are correct, we use the following two conditions:

• \( f(f^{-1}(x)) = x \), for all \( x \) in the domain of \( f^{-1} \)
• \( f^{-1}(f(x)) = x \), for all \( x \) in the domain of \( f \)

Let's look at these verifications with an example using the function \( f(x) = \sqrt{x} \) and its determined inverse \( f^{-1}(x) = x^2 \):

• Substitute \( f^{-1}(x) = x^2 \) into \( f(x) \): \( f(f^{-1}(x)) = \sqrt{x^2} = x \)
• Substitute \( f(x) = \sqrt{x} \) into \( f^{-1}(x) \): \( f^{-1}(f(x)) = (\sqrt{x})^2 = x \)

These verifications confirm that the initial function and its inverse have been accurately calculated and properly inversed.

Square Root Functions

Square root functions are an essential category of mathematical functions characterized by the square root symbol (√), which represents one number whose square is the given number. A basic square root function is \( f(x) = \sqrt{x} \). Understanding this type of function is vital, as they often appear in various areas of math and science.

Some important properties of square root functions include:

• The domain of \( f(x) = \sqrt{x} \) is \( x \geq 0 \), since square roots of negative numbers aren't real in standard math contexts.
• The range is also \( y \geq 0 \), meaning the outputs are non-negative.
• They naturally produce graphics resembling a half-parabola opening to the right.

Square root functions are often involved in solving quadratic equations that can be simplified by removing quadratic terms through inversion, making them an indispensable tool for dealing with equations that involve squares.