Problem 35 Determine if $f$ is one-to-one... [FREE SOLUTION]

91影视

College Algebra with Modeling and Visualization

Gary K. Rockswold

$Math Studyset 91影视 Explanations$ Math

4 Edition

Chapter 5: Problem 35

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

Short Answer

Expert verified

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

Step by step solution

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 $.

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 $[0, \infty)$.

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 $.

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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鈥檚 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.

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91影视

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

Short Answer

Step by step solution

Understand One-to-One Functions

Identify the Domain

Apply the Horizontal Line Test

Verify Algebraically

Key Concepts

Horizontal Line Test

Function Graphing

Square Root Function

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