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Chapter 10: Problem 1

Complete each statement, or answer the question. For a function to have an inverse, it must be ____.

Short Answer

Expert verified
One-to-one (bijective).

Step by step solution

01

Understand the Concept of Function Inverses

To determine if a function has an inverse, remember that an inverse function reverses the operation of the original function. For a function to have an inverse, every output must be attributable to exactly one input.
02

Define the One-to-One Property

A function is one-to-one (bijective) if each element of its domain is mapped to a unique element in its codomain. This means there are no repeated y-values for different x-values.
03

Apply the Horizontal Line Test

A practical way to check if a function is one-to-one is to use the horizontal line test. If every horizontal line crosses the graph of the function at most once, then the function is one-to-one and, therefore, has an inverse.
04

Conclusion

Conclude that for a function to have an inverse, it must be a one-to-one function, meaning it passes the horizontal line test and satisfies the one-to-one property.

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Key Concepts

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

One-to-One Function
A one-to-one function, also known as an injective function, is a function where each element of the domain maps to a unique element in the codomain. This means that no two different inputs can produce the same output.
For example, consider a function \( f(x) = 2x + 3 \). If \( f(x_1) = f(x_2) \), then \( 2x_1 + 3 = 2x_2 + 3 \). Solving this gives \( x_1 = x_2 \), showing that \( f(x) \) is one-to-one because different inputs do not yield the same output.
Recognizing a one-to-one function involves checking that:
  • Each output has one and only one corresponding input.
  • No two distinct inputs share the same output.
Understanding one-to-one functions is essential because they indicate whether a function can have an inverse.
Horizontal Line Test
The horizontal line test is a visual method used to determine whether a function is one-to-one. If every horizontal line crosses the graph of the function at most once, then the function is one-to-one and can have an inverse.
This is because a horizontal line that crosses the graph more than once means there are multiple inputs yielding the same output, which contradicts the one-to-one property.
Here's how to apply the horizontal line test:
  • Draw a series of horizontal lines across the function's graph.
  • Observe each line to ensure it intersects the graph no more than once.
If all horizontal lines meet this condition, the function passes the test, confirming it is one-to-one.
Bijective Function
A bijective function is both one-to-one and onto, making it a powerful type of function in mathematics. Bijective functions have a unique feature: they have inverses.
To be bijective, a function must satisfy two conditions:
  • One-to-One (Injective): Each output is produced from a unique input, as discussed earlier.
  • Onto (Surjective): Every element in the codomain is mapped by an element in the domain.
When a function is bijective, you can confidently determine its inverse because:
  • The inverse will map each element of the codomain back to a unique element of the domain.
  • Every element in the codomain has a preimage in the domain.
Knowing whether a function is bijective helps in various mathematical applications, as it signifies a one-to-one correspondence between the domain and codomain.

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Most popular questions from this chapter

Give the exact real number value of each expression. Do not use a calculator. $$\tan \left(\cos ^{-1} \frac{\sqrt{3}}{2}-\sin ^{-1}\left(-\frac{3}{5}\right)\right)$$

Solve each equation in part (a) analytically over the interval \([0,2 \pi) .\) Then use a graph to solve each inequality in part (b). (a) \(\sin 3 x=0\) (b) \(\sin 3 x<0\)

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal places and approximate values in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\cos x=\sin ^{2} \frac{x}{2}$$

Give the exact real number value of each expression. Do not use a calculator. $$\tan \left(\sin ^{-1} \frac{8}{17}+\tan ^{-1} \frac{4}{3}\right)$$

Write each expression as an algebraic expression in \(u, u>0\). $$\cos \left(\tan ^{-1} \frac{3}{u}\right)$$
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