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Magazine Chapter 5: Problem 9
Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=\sqrt{x} $$
Short Answer
Expert verified
The function \( f(x) = \sqrt{x} \) is one-to-one.
Step by step solution
01
Understand the Function
The function given is \( f(x) = \sqrt{x} \). This is a square root function, which is defined for values of \( x \geq 0 \).
02
Definition of One-to-One Function
A function is said to be one-to-one if every element of the domain maps to a unique element in the range without any repetitions.
03
Test for One-to-Oneness Using the Horizontal Line Test
The horizontal line test states that a function is one-to-one if and only if every horizontal line intersects the graph of the function at no more than one point.
04
Graphing the Function
Graph \( f(x) = \sqrt{x} \). The graph is a curve starting from the origin \((0,0)\) and moving upwards to the right.
05
Applying the Horizontal Line Test
For \( f(x) = \sqrt{x} \), any horizontal line drawn will intersect the graph at most once as the function continuously increases. This satisfies the condition for a one-to-one function.
06
Algebraic Confirmation
To confirm algebraically, assume \( f(a) = f(b) \). This implies \( \sqrt{a} = \sqrt{b} \). Squaring both sides gives \( a = b \), confirming that different inputs result in different outputs. Hence, the function is one-to-one.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Square Root Function
The square root function is represented by the equation \( f(x) = \sqrt{x} \). This function is prevalent in algebra and is defined for non-negative values of \( x \) (i.e., \( x \geq 0 \)).
The output of the square root function increases as the input value of \( x \) increases, but it does so at a decreasing rate. This results in a curve that begins at the origin \((0,0)\) and extends upwards to the right.
Understanding this basic property of the square root function helps in analyzing whether it is one-to-one, as we will explore further using different mathematical approaches.
The output of the square root function increases as the input value of \( x \) increases, but it does so at a decreasing rate. This results in a curve that begins at the origin \((0,0)\) and extends upwards to the right.
Understanding this basic property of the square root function helps in analyzing whether it is one-to-one, as we will explore further using different mathematical approaches.
Horizontal Line Test
The horizontal line test is a simple yet powerful tool to determine if a function is one-to-one. For a function to pass this test, no horizontal line should intersect the graph of the function more than once. If a horizontal line intersects the function at only one point anywhere on its graph, it proves the function is one-to-one.
In the case of the square root function \( f(x) = \sqrt{x} \), its graph rises continuously without falling, meaning any horizontal line will intersect it at most once. This satisfies the condition required for the function to be classified as one-to-one.
Applying the horizontal line test gives us a quick visual confirmation of the one-to-oneness of functions.
In the case of the square root function \( f(x) = \sqrt{x} \), its graph rises continuously without falling, meaning any horizontal line will intersect it at most once. This satisfies the condition required for the function to be classified as one-to-one.
Applying the horizontal line test gives us a quick visual confirmation of the one-to-oneness of functions.
Algebraic Confirmation
Algebraically confirming the one-to-oneness of a function ensures a thorough understanding beyond visual methods. For \( f(x) = \sqrt{x} \), assume two different outputs, \( f(a) \) and \( f(b) \), are equal.
This gives \( \sqrt{a} = \sqrt{b} \).
To solve for \( a \) and \( b \), square both sides to get rid of the square roots: \( a = b \).
This logical reasoning confirms that \( a \) must equal \( b \), meaning different inputs yield different outputs. Thus, algebraically, \( f(x) = \sqrt{x} \) qualifies as a one-to-one function.
This gives \( \sqrt{a} = \sqrt{b} \).
To solve for \( a \) and \( b \), square both sides to get rid of the square roots: \( a = b \).
This logical reasoning confirms that \( a \) must equal \( b \), meaning different inputs yield different outputs. Thus, algebraically, \( f(x) = \sqrt{x} \) qualifies as a one-to-one function.
Function Graphing
Graphing a function helps visualize its behavior and characteristics, such as continuity and the type of curve it forms. For the function \( f(x) = \sqrt{x} \), graphing involves plotting a curve that starts at the origin \((0,0)\) and moves upward to the right, with a shrinking growth rate.
This particular shape illustrates the square root function's characteristic slow growth with increasing \( x \).
By drawing the graph, we can easily apply visual tests like the horizontal line test, thereby reinforcing our understanding of the function's properties, such as it being one-to-one.
This particular shape illustrates the square root function's characteristic slow growth with increasing \( x \).
By drawing the graph, we can easily apply visual tests like the horizontal line test, thereby reinforcing our understanding of the function's properties, such as it being one-to-one.
