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Chapter 1: Problem 81

Determine whether each statement is true or false. If \((0, b)\) is the \(y\) -intercept of a one-to-one function \(f,\) what is the \(x\) -intercept of the inverse \(f^{-1} ?\)

Short Answer

Expert verified
The \( x \)-intercept of the inverse \( f^{-1} \) is \( b \).

Step by step solution

01

Understanding the Relationship

For a one-to-one function \( f \) that has a \( y \)-intercept at \( (0, b) \), we need to understand what this implies. At the \( y \)-intercept, \( x = 0 \), so \( f(0) = b \). This means when \( x \) is 0, \( y \) is \( b \).
02

Exploring the Inverse

The inverse function \( f^{-1} \) swaps the roles of \( x \) and \( y \) in \( f \). This means for every point \( (x, y) \) on \( f \), the corresponding point on \( f^{-1} \) is \( (y, x) \). Therefore, if \( f(0) = b \), then \( f^{-1}(b) = 0 \).
03

Determining the x-intercept of the Inverse

The \( x \)-intercept of \( f^{-1} \) is the \( x \)-value that results in \( y = 0 \), i.e., \( f^{-1}(b) = 0 \). Hence, the \( x \)-intercept of \( f^{-1} \) is \( b \).

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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 is a type of function where each input value has a unique output value. This is an important concept in mathematics because it ensures that every element in the domain maps to a distinct element in the range.
  • This exclusivity means no two different inputs can have the same output.
  • For example, if you have a one-to-one function and in the domain, you plug in two distinct values, say 1 and 2, they should produce distinct results like 3 and 4 in the range.
This uniqueness is critical when considering inverse functions. Only one-to-one functions have inverses that are also functions. The inverse swaps the x and y values for every point, maintaining the uniqueness and allowing the inverse to also follow the function rule.
  • Therefore, for a set to represent an inverse, it must also pass the horizontal line test.
  • In other words, any line crossing the graph of a one-to-one function should intersect the graph at most once.
Y-Intercept
The y-intercept of a function is the point where the graph of the function crosses the y-axis. This point is particularly important because it reveals the output of the function when the input, or x-value, is zero.
In mathematical terms, if the y-intercept of a function is given as
  • (0, b), it means when the input for x is 0, the output y is b.
In the context of a one-to-one function, the y-intercept
  • provides crucial information about how the function behaves, specifically at x=0, lending insight into the function's range at that particular point.
When examining inverse functions, the y-intercept of the original function becomes significant as it swaps place in the coordinate axis. That is, the node (0, b) in function f flips to become (b, 0) in the inverse function.
X-Intercept
The x-intercept of a function is the point where a graph crosses the x-axis, which means the output or y-value is zero.
Understanding this intercept can help you determine when a particular entity, like a projectile in physics or cost in economics, reaches a ground state or a break-even point.
  • Mathematically, it's the input value that makes the output zero.
In the context of inverse functions, finding the x-intercept follows a similar logic.
Given a one-to-one function and its inverse, it is notable that the x-intercept of the inverse function corresponds to the y-intercept of the original function. As described:
  • Since the original function's y-intercept is (0, b), its inverse swaps the coordinates, resulting in the x-intercept being at (b, 0).
  • This shows how swapping the order of coordinates inverts the role of intercepts.

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Most popular questions from this chapter

In Exercises \(39-50\), evaluate \(f(g(1))\) and \(g(f(2)),\) if possible. $$f(x)=\left(1-x^{2}\right)^{1 / 2}, \quad g(x)=(x-3)^{1 / 3}$$

In Exercises \(61-66,\) write the function as a composite of two functions \(f\) and \(g .\) (More than one answer is correct.) $$f(g(x))=\frac{2}{|x-3|}$$

Find the domain of each function, where \(a\) is any positive real number. $$f(x)=-5 \sqrt{x^{2}-a^{2}}$$

In Exercises \(61-66,\) write the function as a composite of two functions \(f\) and \(g .\) (More than one answer is correct.) $$f(g(x))=2(3 x-1)^{2}+5(3 x-1)$$

For Exercises \(95-98,\) refer to the following: In calculus, the difference quotient \(\frac{f(x+h)-f(x)}{h}\) of a function \(f\) is used to find the derivative \(f^{\prime}\) of \(f\) by letting \(h\) approach \(0, h \rightarrow 0\). Find the derivatives of \(F(x)=x, G(x)=x^{2},\) and \(H(x)=(F+G)(x)=x+x^{2} .\) What do you observe?
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