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Chapter 2: Problem 26

Determine whether each function is one-to-one. $$t(x)=\sqrt{x+3}$$

Short Answer

Expert verified
Yes, the function t(x)=鈭�(x+3) is one-to-one.

Step by step solution

01

Understand the Definition of One-to-One Function

A function is one-to-one if every element of the range is mapped by exactly one element of the domain. This means that if for any two different inputs, the outputs are also different, then the function is one-to-one.
02

Apply the One-to-One Condition

For the function t(x)=鈭�(x+3) when replacing any two different inputs, say x_1 and x_2, into the function, should give different outputs if it is one-to-one. (Proof: If t(x_1)=t(x_2) then 鈭�(x_1+3)=鈭�(x_2+3)).
03

Simplify the Equation

If 鈭�(x_1+3)=鈭�(x_2+3), then we square both sides to eliminate the square root: x_1+3=x_2+3
04

Isolate the Variables

Subtract 3 from both sides of the equation to isolate the variables: x_1 = x_2 . This implies that for any two inputs, if the outputs are the same, the inputs must also be the same.

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Key Concepts

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

Function Properties
Understanding function properties is essential to determine if a function is one-to-one. Here are key properties to consider:

  • Input-output relationship: A function maps every input to exactly one output.
  • Unique mapping: In a one-to-one function, each output is the result of exactly one input.
  • Horizontal Line Test: To visually check if a function is one-to-one, you can use the horizontal line test. If no horizontal line intersects the graph of the function more than once, then the function is one-to-one.
For the function in the exercise, we need to check if each input yields a unique output.
Domain and Range
The domain and range of a function help us understand the set of possible inputs and outputs. For the function \(t(x) = \sqrt{x+3}\), let's determine these sets:

  • Domain: The domain is the set of all possible input values (x) that the function can accept. Since the square root function is defined only for non-negative numbers, we have \(x + 3 \geq 0\). Solving this gives \(x \geq -3\). Thus, the domain is \([-3, \infty)\).
  • Range: The range is the set of all possible output values. As a square root function always produces non-negative results, the range of \(t(x)\) is \([0, \infty)\).
Knowing the domain and range is critical to analyzing the function's behavior.
Mathematical Proof
To mathematically prove if \(t(x) = \sqrt{x+3}\) is one-to-one, we follow these steps:

  • Assume Two Equal Outputs: Let \(t(x_1) = t(x_2)\). This gives us \(\sqrt{x_1 + 3} = \sqrt{x_2 + 3}\).
  • Square Both Sides: Squaring both sides eliminates the square root, resulting in \(x_1 + 3 = x_2 + 3\).
  • Isolate the Variable: Subtract 3 from both sides to get \(x_1 = x_2\).
This proof shows that if we get the same output from the function for any two inputs, these inputs must be equal. Thus, \(t(x) = \sqrt{x+3}\) is indeed a one-to-one function.

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