Problem 11 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the function on the given domain. $$ f(x)=2 x^{2}+4,[0, \infty) $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt{\frac{x - 4}{2}} \) with domain \([4, \infty)\).

Step 1: Confirm it's a One-to-One Function

First, we must ensure the function within the given domain is one-to-one to find its inverse. The function \( f(x) = 2x^2 + 4 \) is a parabola that opens upwards. Within the domain \([0, \infty)\), \( f(x) \) is increasing and thus one-to-one. So, we can proceed to find its inverse.

Step 2: Replace \(f(x)\) with \(y\)

Start by writing the equation as \( y = 2x^2 + 4 \). This notation will make it easier to solve for \(x\) in terms of \(y\).

Step 3: Solve the Equation for \(x\)

To find the inverse, solve \( y = 2x^2 + 4 \) for \( x \). Subtract 4 from both sides to get \( y - 4 = 2x^2 \). Divide both sides by 2 to obtain \( x^2 = \frac{y - 4}{2} \). Take the square root of both sides to isolate \( x \). Since the domain is \([0, \infty)\), take the positive square root: \( x = \sqrt{\frac{y - 4}{2}} \).

Step 4: Write the Inverse Function

Replace \( x \) with \( f^{-1}(x) \) and \( y \) with \( x \) in the equation \( x = \sqrt{\frac{y - 4}{2}} \). This gives the inverse function: \( f^{-1}(x) = \sqrt{\frac{x - 4}{2}} \).

Step 5: State the Domain of the Inverse Function

For \( f^{-1}(x) \) to be real and well-defined, the expression under the square root must be non-negative. Therefore, \( x - 4 \geq 0 \), giving \( x \geq 4 \). Thus, the domain of \( f^{-1}(x) \) is \([4, \infty)\).

Key Concepts

One-to-One Functions

A one-to-one function is a function where each output value is uniquely paired with exactly one input value. This means that no two different inputs are mapped to the same output. For a function to have an inverse, it must be one-to-one. This is essential; otherwise, the inverse would not be a function.To determine if a function is one-to-one, you can use the Horizontal Line Test. If any horizontal line intersects the graph of the function at most once, the function is one-to-one. In our exercise, the function is defined as a parabola

• with the equation: \( f(x) = 2x^2 + 4 \)
• restricted to the domain \([0, \infty)\).

Within this domain, the function is increasing, passing the Horizontal Line Test, and therefore one-to-one. This confirmation allows us to find an inverse for the function.

Parabola

A parabola is a symmetrical, curved shape that can open upwards or downwards depending on its equation. In mathematics, a parabola is defined by a quadratic equation of the form \( ax^2 + bx + c \).In the example provided in the exercise, the function is:

• \( f(x) = 2x^2 + 4 \).
• This parabola opens upwards because the coefficient of \( x^2 \) (which is \( 2 \) in this case) is positive.

When talking about a parabola on a restricted domain, as in this exercise with \([0, \infty)\), we are only considering the part of the parabola that stretches out towards the right, continuously increasing. Understanding this behavior is crucial in determining whether the function is one-to-one and, subsequently, finding its inverse.

Function Domain

The domain of a function refers to all the possible input values (\( x \)-values) for which the function is defined. For example, in the exercise, the domain is \([0, \infty)\).This domain is essential because it affects the behavior of the function and whether an inverse can be found.When calculating the inverse of a function, the domain of the original function becomes the range of its inverse.

• In the original function, the domain is \([0, \infty)\).
• Therefore, the inverse function will have this range.

The process of determining the inverse function also involves identifying the domain of the inverse. In our exercise, we learned that for the inverse function \( f^{-1}(x) \) to be valid, the expression inside the square root (\( x - 4 \)) must be non-negative. This gives us the domain \([4, \infty)\) for the inverse function.

Square Root

The square root operation, denoted as \( \sqrt{\cdot} \), finds the number that, when squared, yields the original value. It is a fundamental concept in algebra, and it appears in the solution as the step necessary to solve for \( x \) when finding the inverse.In the given function, to isolate \( x \) we end up with:

• \( x^2 = \frac{y-4}{2} \).

Taking the square root of both sides, you derive:

• \( x = \sqrt{\frac{y - 4}{2}} \).

The important part to remember is that since we're dealing with the inverse of a function with a domain of \([0, \infty)\), we take the positive square root. This ensures the result is a valid function and retains the integrity of the initial function's domain restrictions. Watch out for negative numbers inside a square root, as the operation doesn’t process them in the realm of real numbers.