Problem 35 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=x^{1 / 2} $$

Step-by-Step Solution

Answer

The function \( f(x) = x^{1/2} \) is one-to-one on its domain \([0, \infty)\).

Step 1: Understand One-to-One Functions

A function is one-to-one if each output value corresponds to exactly one input value, meaning no two different inputs have the same output. Mathematically, we say a function \( f \) is one-to-one if \( f(a) = f(b) \) implies \( a = b \).

Step 2: Identify the Domain

The function \( f(x) = x^{1/2} \) is defined for all \( x \geq 0 \) as it represents the square root function. This means its domain is \([0, \infty)\).

Step 3: Apply the Horizontal Line Test

The horizontal line test states that a function is one-to-one if no horizontal line intersects the graph of the function more than once. Considering \( f(x) = x^{1/2} \), graph the function. Notice that for each value of \( y \), corresponding to \( f(x) \), there is only one value of \( x \).

Step 4: Verify Algebraically

To verify if \( f(x) = x^{1/2} \) is one-to-one, assume \( f(a) = f(b) \), which implies \( a^{1/2} = b^{1/2} \). Squaring both sides gives \( a = b \), confirming no two distinct numbers will have the same square root.

Key Concepts

Horizontal Line Test

To determine if a function is one-to-one, we often use the horizontal line test. This test is simple yet powerful for identifying functions that are one-to-one, as it graphically represents whether each output corresponds to only one input.

Here’s how it works:

• Draw the graph of the function on a coordinate plane.
• Imagine or use a ruler to draw horizontal lines (parallel to the x-axis) across the graph.
• If any horizontal line intersects the function's graph more than once, the function is not one-to-one.
• If each horizontal line touches the function's graph exactly once, the function is one-to-one.

This test helps visualize the relationship between inputs and outputs. In the case of the function \( f(x) = x^{1/2} \), drawing the graph and applying the horizontal line test shows that no horizontal line will intersect the graph more than once, indicating the function is one-to-one within its domain.

Function Graphing

Function graphing is a fundamental skill that helps in understanding the behavior of functions visually. By graphing a function, we get insights that purely numerical or algebraic descriptions might not immediately reveal.

For the function \( f(x) = x^{1/2} \), start by recognizing it as the square root function. Here’s how to graph the function:

• Identify the domain of the function \([0, \infty)\).
• Plot key points: For example, \( (0,0) \), \( (1,1) \), and \( (4,2) \).
• Draw a smooth curve through these points, recognizing that the function increases gradually.
• Note that the graph exists only in the first quadrant, as the function is undefined for negative \( x \).

Graphing helps in understanding how the function behaves and assists in the visual application of the horizontal line test.

Square Root Function

The square root function is a special type of function with unique properties that are important in mathematics. This function is defined as \( f(x) = \sqrt{x} \) or equivalently \( f(x) = x^{1/2} \).

Key properties of the square root function include:

• Domain: The domain of \( \sqrt{x} \) is all non-negative numbers; thus, \( x \geq 0 \).
• Range: The output is also non-negative, \( y \geq 0 \).
• Behavior: The function is defined only in the first quadrant and it increases gradually without bound as \( x \) grows larger.
• One-to-One Nature: Since each \( y \) value corresponds to a unique \( x \), it demonstrates that the function is one-to-one.
• Simplification: Algebraically, if \( \sqrt{a} = \sqrt{b} \), then \( a = b \), which supports the conclusion of it being one-to-one.

Understanding the square root function is crucial for more advanced topics in algebra and calculus. It's a great example of how specific behaviors are restricted due to the nature of the function's definition.