Problem 19 - College Algebra

Question

For the following exercises, use a graphing utility to determine whether each function is one-to-one. $$ f(x)=\sqrt{x} $$

Step-by-Step Solution

Answer

Yes, the function \( f(x) = \sqrt{x} \) is one-to-one.

Step 1: Understand the Definition of a One-to-One Function

A function is considered one-to-one if each output value is associated with exactly one input value—that is, no two different inputs map to the same output. Graphically, this means that a horizontal line should intersect the graph of the function at most once.

Step 2: Analyze the Function Graph

The given function is \( f(x) = \sqrt{x} \). It's important to know that its graph is a curve starting at the origin, going towards the right, and continuously increasing. To determine if it's one-to-one, we'll analyze its graph using a graphing utility.

Step 3: Use a Horizontal Line Test

Utilize a graphing utility to plot the graph of \( f(x) = \sqrt{x} \). Observe how any horizontal line (parallel to the x-axis) intersects the graph. Since each horizontal line will intersect the graph at most once, this implies that no two different inputs share the same output.

Step 4: Conclude the Function's Properties

Upon using the graphing utility, observe and confirm that every horizontal line across the domain of \( f(x) \) intersects the graph at most once. Therefore, the function \( f(x) = \sqrt{x} \) is one-to-one.

Key Concepts

Horizontal Line Test

The horizontal line test is a simple but powerful tool used to determine if a function is one-to-one. It's a geometrical method that helps visualize whether any horizontal line (parallel to the x-axis) can intersect a graph more than once. If a horizontal line cuts the graph more than once, the function is not one-to-one.

For the function \( f(x) = \sqrt{x} \), imagine drawing horizontal lines across its graph—these lines run parallel to the x-axis. You will notice that each line crosses the curve at most at one point.

• This means there are no two different 'x' values that give the same 'y' value.
• If a function passes this test, it's confirmed to be a one-to-one function.

This is why the horizontal line test is fundamental to understanding and confirming the one-to-one nature of functions, as it provides a clear visual affirmation.

Graphing Functions

Graphing functions allows us to see their behavior visually, making it easier to determine properties like being one-to-one. For \( f(x) = \sqrt{x} \), its graph starts at the origin (0,0) and moves upward to the right, illustrating a continuously increasing pattern.

Using a graphing utility can help you plot \( f(x) = \sqrt{x} \) accurately, providing you with an interactive visual of the function's behavior.

• Graphs give a quick insight into whether the horizontal line test will pass.
• A well-plotted graph not only shows the function's direction but also displays its range and domain clearly.

This visual representation is crucial because it allows you to see clearly whether the function maintains uniqueness in terms of inputs leading to outputs.

Function Properties

Understanding function properties is key to analyzing mathematical functions thoroughly. A one-to-one function has a special property where each input corresponds to a unique output, with no repeats allowed for y-values from different x-values.

For the function \( f(x) = \sqrt{x} \), its properties include:

• Domain: Non-negative real numbers \( x \geq 0 \), because you cannot take the square root of negative numbers in real terms.
• Range: Non-negative real numbers \( y \geq 0 \), reflecting the fact that square roots yield non-negative results.
• Monotonic Behavior: The function is strictly increasing, meaning that as \( x \) increases, \( f(x) \) increases without any decrease—that's a characteristic that supports its one-to-one nature.

Understanding these properties provides a deeper insight into why the horizontal line test works and why the function is one-to-one. These properties form the backbone of analyzing and sketching any function accurately.