Problem 32 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Function Type

The given function is a quadratic function, specifically in the form of \( f(x) = 4 - x^2 \). Since the domain is restricted to \( x \geq 0 \), we know we are dealing with only a portion of the full parabola.

Step 2: Swap Variables for Inversion

To find the inverse, we start by swapping \( x \) and \( y \) in the equation. Since our function is \( y = 4 - x^2 \), the swapped equation is \( x = 4 - y^2 \).

Step 3: Solve for the New 'y'

To isolate \( y \), rearrange the equation \( x = 4 - y^2 \) as follows:1. Subtract 4 from both sides: \( x - 4 = -y^2 \).2. Multiply each side by -1: \( -(x - 4) = y^2 \) or \( y^2 = 4 - x \).3. Take the square root of both sides to solve for \( y \): \( y = \sqrt{4-x} \).

Step 4: Determine Domain and Range for Inverse

Given \( f(x) = 4 - x^2 \) has a domain of \( x \geq 0 \), the range is derived as \( 0 \leq f(x) \leq 4 \) since the function decreases from \( 4 \) to \( 0 \). For the inverse function \( f^{-1}(x) = \sqrt{4-x} \), the domain is \( 0 \leq x \leq 4 \) and the range is \( f^{-1}(x) \, \geq \, 0 \).

Step 5: Graph the Original and Inverse Functions

Plot the original function \( f(x) = 4 - x^2 \) using its domain \( x \geq 0 \), which is a downward-opening parabola starting at \( (0, 4) \) and ending at the x-intercept \( (2, 0) \). The inverse function \( f^{-1}(x) = \sqrt{4-x} \) is a half-parabola starting at \( (0, 2) \) and ending at \( (4, 0) \). The line \( y = x \) can be used to verify the correct reflection of these graphs.

Key Concepts

Quadratic Functions

A quadratic function is a type of polynomial function characterized by the presence of an exponent of two on the variable. The general form of a quadratic function is \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \).Quadratic functions graph as parabolas that either open upwards or downwards, depending on the sign of \( a \). If \( a > 0 \), the parabola opens upward. If \( a < 0 \), it opens downward.
In our given exercise, the function \( f(x) = 4 - x^2 \) is a specific type of quadratic function with \( a = -1 \), indicating a downward-opening parabola. The vertex form of this function reveals that its vertex is at the point \( (0, 4) \), giving it maximum value at the vertex within the restricted domain \( x \geq 0 \). The restriction ensures that only the portion beginning at the vertex continues towards the right, consistent with the range {0, 4}.
Recognizing the functional properties of quadratic equations helps us effectively handle inversions, because we are primarily concerned with identifying a specific segment of the parabola when the domain is constrained.

Graphing Functions

When graphing functions, such as the one in our exercise, we utilize the function's properties to create a visual representation on the coordinate plane. For the quadratic function \( f(x) = 4 - x^2 \) where \( x \geq 0 \), we note that it forms a curve known as a parabola.
Steps in graphing:

• Identify the vertex and intercepts. Here, the vertex is at \( (0, 4) \) and it intercepts the x-axis at \( (2,0) \).
• Since the parabola opens downward, it extends from the vertex down towards the x-axis.
• For the inverse, the swapped equation \( f^{-1}(x) = \sqrt{4-x} \) is graphed. This graph starts at \( (0, 2) \) and decreases toward \( (4, 0) \).


• To check the correctness, plot the line \( y = x \) which serves as a line of symmetry between the function and its inverse graph.

A visual graph aids in understanding the relationship between a function and its inverse.It shows us the reflection across the line \( y = x \), reinforcing the inverse concept.

Domain and Range of Functions

Domain and range are fundamental concepts in understanding functions. The domain represents all possible inputs (x-values) a function can accept, while the range consists of all possible outputs (y-values) generated by the function.
In the initial function, \( f(x) = 4 - x^2 \), we see the domain is restricted to non-negative \( x \)-values \( (x \geq 0) \). This means we are only considering the right half of the parabola. This portion of the parabola solves for \( 0 \leq f(x) \leq 4 \), indicating the function's outputs between 0 and 4.
When we obtain the inverse, \( f^{-1}(x) = \sqrt{4-x} \), the domain changes to \( 0 \leq x \leq 4 \), and the range is derived to be \( f^{-1}(x) \geq 0 \). This illustrates how inversing affects domain and range while maintaining continuity between the inverses. The original function's range becomes the inverse function's domain and vice-versa.
Understanding these relationships is crucial when dealing with transformations and inversions of functions, providing insight into how constraints are redefined through the inversion process.