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Chapter 3: Q5E (page 161)

Determine whether the function given by a graph is one-to-one.

Short Answer

Expert verified

The resultant answer is not a one-to-one function.

Step by step solution

01

Given data

The given function \(f\) is one-to-one.

02

Concept of functions

The simplest definition is an equation will be a function if, for any \({\rm{x}}\) in the domain of the equation (the domain is all the \({\rm{x}}\)'s that can be plugged into the equation), the equation will yield exactly one value of \({\rm{y}}\) when we evaluate the equation at a specific \({\rm{X}}\).

03

Simplify the expression

Horizontal line test states that the graph of the function is one-to-one function if and only if a horizontal line intersects the graph exactly once.

Perform the horizontal line test for the given graph.

Draw a horizontal line such that it passes through the curve as shown below.

From Figure, it is observed that the horizontal line intersects the curve at two distinct points, which means it fails the horizontal line test.

Therefore, the function given by a graph is not a one-to-one function.

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Most popular questions from this chapter

To prove the identity \(cosh( - x) = coshx\).

To determine the value of \(\mathop {\lim }\limits_{x \to {2^ - }} \left( {{e^{\frac{3}{{2 - x}}}}} \right)\).

Sketch the graphs of the function \(y = {e^x},y = {e^{ - x}},y = {8^x}\)and \(y = {8^{ - x}}\) on the same axes and interpret how these graphs are related.

(a) To show the function\(f(x) = {x^3}\)is one-to-one.

(b) The value of\({\left( {{f^{ - 1}}} \right)^\prime }(8)\), where\(f(x) = {x^3}\).

(c) The inverse of the function\(f(x) = {x^3}\)and state its domain and range.

(d) Whether the value of\(\left( {{f^{ - 1}}} \right)(8)\)is\(\frac{1}{{12}}\)by using the inverse function.

(e) To sketch: The graph of\(f(x) = {x^3}\)and\({f^{ - 1}}(x) = \sqrt(3){x}\)in the same coordinate axis.

(A) to determine the function\(f(x) = \frac{1}{{x - 1}},\;x > 1\)is one-to-one.\({f^{ - 1}}(x) = \frac{1}{x} + 1\)in

(B) To determine the value of\({\left( {{f^{ - 1}}} \right)^\prime }(2)\), where\(f(x) = \frac{1}{{x - 1}}\).

(C) To determine the inverse of the function\(f(x) = 9 - {x^2}\)and state its domain and range.

(D) To determine whether the value of\({\left( {{f^{ - 1}}} \right)^\prime }(2)\)is\(\frac{1}{4}\)using the inverse function.

(E) To sketch: The graph of\(f(x) = \frac{1}{{x - 1}}\)and\({f^{ - 1}}(x) = \frac{1}{x} + 1\)in the same coordinate.
See all solutions

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