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Chapter 2: Problem 3

If (1,10) lies on the graph of \(y=f(x),\) what can be said about the graph of \(y=f^{-1}(x) ?\)

Short Answer

Expert verified
The point (10,1) lies on the graph of \(y=f^{-1}(x)\).

Step by step solution

01

Understand the Relationship between Functions and Their Inverses

The inverse function, denoted as \(f^{-1}(x)\), is a function that undoes the operation of the \(f(x)\) function. It is derived from the original function in such a way that if \((a, b)\) is a point on the graph of \(y = f(x)\), then \((b, a)\) will be a point on the graph of \(y = f^{-1}(x)\).
02

Identify the Given Point on the Original Graph

We are given that the point \((1, 10)\) lies on the graph of \(y = f(x)\). This means that when \(x = 1\), the output of the function is \(y = 10\).
03

Determine the Corresponding Point on the Inverse Graph

By the property of inverse functions, the roles of \(x\) and \(y\) are switched. Therefore, if \((1, 10)\) is on \(y = f(x)\), then \((10, 1)\) is on \(y = f^{-1}(x)\), as these points are reflections of each other along the line \(y = x\).

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Key Concepts

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

Function Graph
In mathematics, a function graph visually represents the relationship between inputs and outputs of a function. It's like a map showing all the ordered pairs \(x, y\) where \(y = f(x)\). Each point on this graph corresponds to a solution of the function equation, linking an input value \(x\) to a single output value \(y\).
Visualizing function graphs is an invaluable tool in understanding how the function behaves and how it transforms inputs to outputs. For instance:
  • They can reveal trends, like whether the function is increasing or decreasing.
  • They can show important features, such as intercepts, where the function crosses the axes.
  • They demonstrate continuity, indicating how smoothly the function behaves between points.
Understanding these properties can help unpack the deeper characteristics of a function's behavior, making it simpler to analyze and apply.
Inverse Graph
An inverse graph represents the inverse function \(f^{-1}(x)\), which swaps the roles of inputs and outputs from the original function \(f(x)\). If \(f(x)\) transforms \(x\) into \(y\), then \(f^{-1}(x)\) transforms \(y\) back into \(x\). This swapping effectively "undoes" the operations of the original function. For example:
  • A point \( (a, b) \) on the graph of a function becomes \( (b, a) \) on the graph of its inverse.
  • The inverse graph is a reflection of the original graph about the line \( y = x \).
  • This reflection can be visualized as folding the plane along the line \( y = x \) where the original and inverse graphs meet.
This swap in roles between \(x\) and \(y\) is crucial in solving equations and understanding the relationships in mathematics, allowing for comprehensive treatment of mathematical problems involving inverses.
Coordinate Reflection
Coordinate reflection is a concept where points on a graph are mirrored with respect to some line or axis. In the context of inverse functions, this reflection occurs over the line \( y = x \). This works as follows:
  • For a point \( (a, b) \) on the function graph \(y = f(x)\), there is a corresponding point \( (b, a) \) on the inverse graph \(y = f^{-1}(x)\).
  • The reflection over \( y = x \) results in a switch between \(x\) and \(y\) coordinates, creating this mirroring effect.
  • This means that what was originally an input becomes an output and vice versa.
Understanding coordinate reflection is vital, especially when visually identifying inverse functions. This mirroring property not only aids in graphically identifying inverses, but also in verifying algebraic work and understanding mathematical symmetries.

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Most popular questions from this chapter

Compute the derivative of the given function. $$f(x)=\sqrt{x}+\frac{1}{\sqrt{x}}$$

Compute the derivative of the given function. $$f(x)=\sqrt[3]{x}+x^{2 / 3}$$

Find \(\frac{d y}{d x}\) using implicit differentiation. $$\left(y^{2}+2 y-x\right)^{2}=200$$

Find the equation of the tangent line to the graph of the implicitly defined function at the indicated points. As a visual aid, each function is graphed. \(\left(x^{2}+y^{2}+x\right)^{2}=x^{2}+y^{2}\) (a) \(\operatorname{At}(0,1)\) (b) At \(\left(-\frac{3}{4}, \frac{3 \sqrt{3}}{4}\right)\).

Use logarithmic differentiation to find \(\frac{d y}{d x}\), then find the equation of the tangent line at the indicated \(x\) -value. $$y=\frac{x^{x}}{x+1}, \quad x=1$$
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