Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$ f(x)=4-x^{2}, \quad x \geq 0 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt{4 - x} \).

Step 1: Identify the Function and Its Domain

The given function is \( f(x) = 4 - x^2 \) with the domain \( x \geq 0 \), which means that only non-negative values of \( x \) are considered.

Step 2: Swap the Variables

To find the inverse, start by swapping \( x \) and \( y \) in the equation. So, replace \( f(x) \) with \( y \) to get \( y = 4 - x^2 \), and then write it as \( x = 4 - y^2 \).

Step 3: Solve for y

Solve the equation \( x = 4 - y^2 \) for \( y \). Rearrange it to \( y^2 = 4 - x \).

Step 4: Take the Square Root

To find \( y \), take the square root of both sides: \( y = \sqrt{4 - x} \). Since the original function's input \( x \geq 0 \), the output \( y \) should also be \( y \geq 0 \).

Step 5: Write the Inverse Function

The inverse function is \( f^{-1}(x) = \sqrt{4 - x} \) with the domain \( 0 \leq x \leq 4 \).

Step 6: Graph the Functions

Plot both the original function \( f(x) = 4 - x^2 \) for \( x \geq 0 \) and its inverse \( f^{-1}(x) = \sqrt{4 - x} \). The inverse will be a reflection of the original across the line \( y = x \).

Key Concepts

Function Graphing

Graphing a function is a key method for visually representing a mathematical relationship. To graph a function, you need to plot points on a coordinate plane where the x-axis represents the input and the y-axis represents the output. This gives us a clear picture of the function's behavior, such as whether it is increasing, decreasing, or constant over its domain. When graphing both a function and its inverse, it's helpful to note that the inverse can be seen as a reflection of the original function across the line \( y = x \). This means that if you were to fold your graph along this line, the function and its inverse would align perfectly. To ensure accuracy, you might:

• Choose several values of \( x \) within the domain.
• Calculate the corresponding \( y \) values using the function.
• Plot these \( (x, y) \) pairs on the graph.
• Repeat the process for the inverse function.

The intersection line \( y = x \) helps verify accuracy, providing a reference point for reflecting both graphs.

Domain and Range

Understanding a function's domain and range helps define where the function "lives." The domain of a function includes all possible input values (x-values) for which the function is defined. Conversely, the range is the set of output values (y-values) that the function can produce. For the function \( f(x) = 4 - x^2 \), the domain is \( x \geq 0 \), meaning it only includes non-negative x-values. This restriction is crucial since without it, the inverse function wouldn't exist as a true function. This constraint ensures that each input has a single unique output, essential for function inverses. The range for \( f(x) \) is from 0 to 4, inclusive, because when \( x \) is 0, \( y = 4 \), and when \( x \) is at its maximum of 2, \( y \) becomes 0.With the inverse function, \( f^{-1}(x) = \sqrt{4-x} \), the domain flips to being 0 to 4 since the inverse uses the original range as its domain. Meanwhile, its range becomes \( y \geq 0 \), as dictated by the original's domain.

Quadratic Functions

Quadratic functions are a specific type of polynomial function characterized by their highest degree term being squared, typically taking the form \( ax^2 + bx + c \). The function \( f(x) = 4 - x^2 \) is quadratic, distinguished by having one squared term and a constant.Quadratic functions often create a parabolic graph. Here, since \( a \) is negative (-1 in front of \( x^2 \)), this particular parabola opens downwards. The vertex, which is the peak or trough of the parabola, is a critical point, determining its symmetry and maximum or minimum value.In \( f(x) = 4 - x^2 \):

• The vertex is \((0, 4)\), signaling the highest point on the graph.
• The function decreases as \( x \) progresses positively from zero, reflecting the downwards slope.

Grasping the nature of quadratic functions involves understanding their standard features and how they apply to specific forms, such as factoring, completing the square, or employing the quadratic formula. Each technique provides different pathways to interpret and solve quadratic equations.