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Chapter 7: Problem 60

For each function: $$f(x)=\sqrt{x+2}$$ a) Graph the function. b) Determine whether the function is one-to-one. c) If the function is one-to-one, find an equation for its inverse. d) Graph the inverse of the function.

Short Answer

Expert verified
The function \(f(x) = \sqrt{x + 2}\) is one-to-one. Its inverse is \(f^{-1}(x) = x^2 - 2\).

Step by step solution

01

- Graph the function

To graph the function, identify key points. For \(f(x) = \sqrt{x + 2}\), the domain is \(x \geq -2\). Plot points like \(x = -2\), \(x = 0\), and \(x = 2\) and draw the curve.
02

- Determine if the function is one-to-one

A function is one-to-one if it passes the Horizontal Line Test (no horizontal line intersects the graph more than once). Since \(f(x)\) is strictly increasing over its domain, it is one-to-one.
03

- Find the inverse of the function

For the function \(f(x) = \sqrt{x + 2}\), swap \(x\) and \(y\) to find the inverse: \(x = \sqrt{y + 2}\). Solving for \(y\) gives \(y = x^2 - 2\). So, the inverse is \(f^{-1}(x) = x^2 - 2\).
04

- Graph the inverse function

Using the inverse function \(f^{-1}(x) = x^2 - 2\), plot points like \(x = 0\), \(x = 1\), and \(x = -1\) and draw the parabola. Note its domain \(x \geq 0\) ensures the original function's range matches.

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Key Concepts

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

One-to-One Functions
A one-to-one function is a type of function where each input produces a unique output. This means no two different inputs will map to the same output value. An easy way to understand this is to think about how each key on a keyboard is mapped to a different letter: each key (input) only produces one specific letter (output).
If a function is one-to-one, it guarantees that an inverse function exists. This uniqueness is crucial because, without it, an inverse relationship couldn't be established in a meaningful way. To check if the function is one-to-one, you can use the Horizontal Line Test.
Horizontal Line Test
The Horizontal Line Test is a straightforward method to determine if a function is one-to-one. By drawing horizontal lines (parallel to the x-axis) across the graph of a function:
  • If any horizontal line intersects the graph more than once, the function is not one-to-one.
  • If every horizontal line intersects the graph at most once, the function is one-to-one.

For instance, in the given function, \( f(x) = \sqrt{x + 2} \), every horizontal line will intersect the curve exactly once, showing the function is indeed one-to-one.
Graphing Functions
Graphing functions involves plotting key points on a coordinate plane to represent the relationship between the input (x-values) and the output (y-values). For the function \( f(x) = \sqrt{x + 2} \), follow these steps:
  • Identify the domain and range. The domain here is \( x \geq -2 \) (because the square root function requires non-negative inputs).
  • Choose key points within the domain, such as \( x = -2, 0, 2 \).
  • Calculate corresponding y-values using the function, yielding points such as \( (-2, 0), (0, \sqrt{2}), (2, 2) \).
  • Plot these points on the graph and draw a smooth curve connecting them.

Similarly, for inverse functions like \( f^{-1}(x) = x^2 - 2 \), plot relevant points within the new domain and connect them to visualize the inverse.

Graphing gives a visual representation and helps in better understanding function behavior and characteristics.

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Most popular questions from this chapter

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