Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 17-20: Use the theorem on inverse functions to prove that \(f\) and \(g\) are inverse functions of each other, and sketch the graphs of \(f\) and \(g\) on the same coordinate plane. $$ f(x)=-x^{2}+3, x \geq 0 ; \quad g(x)=\sqrt{3-x}, x \leq 3 $$

Step-by-Step Solution

Answer

Functions \( f \) and \( g \) are indeed inverses of each other.

Step 1: Define the theorem

The theorem on inverse functions states that if a function \( f \) is invertible, then \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \) in the domains of \( g \) and \( f \), respectively. We need to verify these conditions for \( f(x) = -x^2 + 3 \) and \( g(x) = \sqrt{3-x} \).

Step 2: Verify \(f(g(x)) = x\)

Substitute \( g(x) \) into \( f \):\[ f(g(x)) = f(\sqrt{3-x}) = - (\sqrt{3-x})^2 + 3 \]Simplify:\[ = -(3-x) + 3 \]\[ = x \]The equation \( f(g(x)) = x \) is satisfied for \( x \leq 3 \).

Step 3: Verify \(g(f(x)) = x\)

Substitute \( f(x) \) into \( g \):\[ g(f(x)) = g(-x^2 + 3) = \sqrt{3 - (-x^2 + 3)} \]Simplify:\[ = \sqrt{x^2} \]\[ = x \quad (\text{since} \; x \geq 0) \]The equation \( g(f(x)) = x \) is satisfied for \( x \geq 0 \).

Step 4: Graph the functions

Graph \( f(x) = -x^2 + 3 \) and \( g(x) = \sqrt{3-x} \) on the same coordinate plane. Note that \( f \) is a downward-opening parabola with vertex at \( (0, 3) \), restricted to \( x \geq 0 \), and \( g \) is a decreasing half of a horizontal parabola, restricted to \( x \leq 3 \). Both functions will be reflections of each other across the line \( y = x \).

Key Concepts

Graphs of Functions

Understanding the graphs of functions provides a visual insight into the relationship between the input and output values of the function. For the given functions, we have:

• Function \( f(x) = -x^2 + 3 \) represents a downward-opening parabola. Since it's restricted to \( x \geq 0 \), only the right side of the parabola is graphed.
• Function \( g(x) = \sqrt{3-x} \) is the inverse, and it represents a curve that forms half of a horizontal parabola, restricted to \( x \leq 3 \).

Graphing both functions on the same coordinate plane reveals that they are symmetrical about the line \( y = x \). This symmetry is a characteristic property of inverse functions. The reflection around this line emphasizes how the values of \( f(x) \) and \( g(x) \) swap roles between the x-axis and y-axis.

Theorem on Inverse Functions

The theorem on inverse functions is a foundational concept in determining whether two functions are inverses of one another. It asserts that two functions \( f \) and \( g \) are inverses if they satisfy both \( f(g(x)) = x \) and \( g(f(x)) = x \) for all relevant values of \( x \). These relationships reveal:

• For \( f \) and \( g \) to be inverse, every input \( x \) into \( g \) that is then fed into \( f \) should return \( x \), and vice versa.
• This theorem confirms that the composition of a function and its inverse results in the identity function, denoted by \( x \).

Let's see how it applies to the functions addressed here: For \( f(x) = -x^2 + 3 \) and \( g(x) = \sqrt{3-x} \), both \( f(g(x)) = x \) and \( g(f(x)) = x \) are proven correct. Thus, they indeed are inverse functions.

Function Composition

Function composition is when the output from one function becomes the input for another. In the case of inverse functions, composing them simplifies to the identity function \( x \):

• For \( f(g(x)) \), function \( g(x) = \sqrt{3-x} \) is substituted into \( f(x) \), resulting in simplification back to \( x \), given \( x \leq 3 \).
• Similarly, for \( g(f(x)) \), using \( f(x) = -x^2 + 3 \) as input into \( g(x) \) simplifies to \( x \), considering \( x \geq 0 \).

This replacement reinforces why inverse functions can "undo" each other, reflecting the fundamental nature of these pairs of functions. It also stresses the interplay between functions, encouraging a clear check whether two functions are truly inverses.

Domain and Range

Domains and ranges play an essential role in understanding and defining functions and their inverses.

• The domain of a function is the set of input values for which the function is defined. For \( f(x) = -x^2 + 3 \), the domain is \( x \geq 0 \), and for \( g(x) = \sqrt{3-x} \), it is \( x \leq 3 \).
• The range is the set of output values; for \( f(x) \), the range is \( y \leq 3 \), while for \( g(x) \), it is \( y \geq 0 \).

These restrictions define where each function can "live" on a coordinate plane, and importantly, they ensure the inverse relationship holds by confining each function to values where the theorem on inverse functions can be applied effectively. Understanding these aspects is crucial for determining the suitability of inverse functions in different contexts.