Problem 22 - Precalculus

Question

With help from your classmates, find the inverses of the functions in Exercises \(21-24\). $$f(x)=a \sqrt{x-h}+k, a \neq 0, x \geq h$$

Step-by-Step Solution

Answer

The inverse of the function is \( f^{-1}(x) = \left( \frac{x - k}{a} \right)^2 + h \).

Step 1: Understand the Function

The given function is in the form \( f(x) = a \sqrt{x-h} + k \), where \( a eq 0 \). The domain is restricted to \( x \geq h \) to ensure the square root is defined with real numbers.

Step 2: Express the Function in Terms of y

Set \( y = f(x) \). This gives us \( y = a \sqrt{x-h} + k \), which we need to solve for \( x \) as a function of \( y \).

Step 3: Isolate the Square Root

Subtract \( k \) from both sides to get \( y - k = a \sqrt{x-h} \).

Step 4: Solve for the Square

Divide both sides by \( a \) to isolate the square root: \( \frac{y - k}{a} = \sqrt{x-h} \).

Step 5: Eliminate the Square Root

Square both sides of the equation to eliminate the square root: \( \left( \frac{y - k}{a} \right)^2 = x - h \).

Step 6: Solve for x

Add \( h \) to both sides to solve for \( x \): \( x = \left( \frac{y - k}{a} \right)^2 + h \).

Step 7: Write the Inverse Function

Swap \( x \) and \( y \) to get the inverse function: \( f^{-1}(x) = \left( \frac{x - k}{a} \right)^2 + h \), ensuring \( y - k > 0 \).

Key Concepts

Understanding Function Transformation

Function transformation is a concept that involves altering the appearance or position of a function's graph without changing its shape. In the case of the function \( f(x) = a \sqrt{x-h} + k \), we explore several transformation types:

1. **Vertical Transformation:** The parameter \( k \) adds or subtracts from the function's output, shifting the graph vertically. If \( k \) is positive, the graph moves upward. If \( k \) is negative, it shifts downward.

2. **Horizontal Transformation:** The parameter \( h \) shifts the graph horizontally. When \( h \) is positive, the graph moves to the right; when it's negative, the graph shifts to the left.

3. **Vertical Stretch/Compression:** The coefficient \( a \) affects the graph's steepness. If \( \lvert a \rvert > 1 \), the graph stretches vertically, and if \( 0 < \lvert a \rvert < 1 \), it compresses.

These transformations help in visualizing the changes applied to the standard square root function, creating a variety of new graphs while retaining the core characteristics of the original function.

The Nature of Square Root Functions

Square root functions are fundamental mathematical expressions involving the square root of a variable. A basic square root function is \( f(x) = \sqrt{x} \). In transformations, we modify this to \( f(x) = a \sqrt{x-h} + k \), reflecting several modifications through an applied formula.

**Characteristics of the Square Root Function:**
1. **Domain:** The domain is defined as all real numbers \( x \) for which the expression under the square root is non-negative. For \( f(x) = \sqrt{x} \), it's \( x \geq 0 \).
2. **Range:** For a standard \( f(x) = \sqrt{x} \), the range is also all non-negative real numbers.

• **Graph Behavior:** The graph of a basic square root function starts at the origin and rises steadily to the right.

In transformed versions, understanding how parameters modify the domain and position is crucial. The transformations revolve around keeping the square root function valid while changing its basic structure through coefficients and additional constants.

Exploring Function Domain

The domain of a function is the set of all possible input values \( x \) that make the function real and valid. For the function \( f(x) = a \sqrt{x-h} + k \), the expression under the square root \( x-h \) must be non-negative to ensure we have real number outputs.

**Determining the Domain:**

• For the function to be defined, \( x-h \geq 0 \).
• Therefore, we find \( x \geq h \) is the required domain condition.
• This limits the function to operate only within these inputs, ensuring the validity of the square root.

Understanding domain constraints is essential because it prevents errors in calculation and application. The domain restriction \( x \geq h \) directly influences how the graph can appear and the real numbers we can use within operations.

The Importance of Real Numbers in Functions

Real numbers play a critical role in functions, as they represent a complete, continuous set of numbers, including all rational and irrational numbers. These numbers are essential for defining function domains and ranges.

In the function \( f(x) = a \sqrt{x-h} + k \):

• **Real Outputs:** We require that the result inside the square root is a real number, hence conditions like \( x \geq h \) to maintain this feasibility.
• **Comprehensive Graphing:** Real numbers allow the function to be graphically represented, spanning from negative to positive infinities (where applicable within domain constraints).

Values in a function that are not real numbers (such as imaginary numbers from negative square roots) would make the function undefined. Thus, understanding and applying real numbers correctly ensures the function remains a valid and useful mathematical tool.