Question

1. Determine the one-to-one function.
2. Determine the graph of the function is one-to-one.

Short answer

(a) The resultant answer is\(f\left( {{x_1}} \right) \ne f\left( {{x_2}} \right){\kern 1pt} {\kern 1pt} \)whenever \({\kern 1pt} {x_1} \ne {x_2}{\kern 1pt} \)or \(f\left( {{x_1}} \right) = f\left( {{x_2}} \right){\kern 1pt} {\kern 1pt} {\kern 1pt} \) whenever \({x_1} = {x_2}\).

(b) A graph is one-to-one if and only if the graph of \(f\) intersects the horizontal line at most once.

Step-by-step solution

Given data

The given function is one-to-one.

Concept of limits

Limits concept is based on the idea of closeness, used primarily to assign values to certain functions at points where no values are defined, in such a way as to be consistent with nearby values.

Simplify the expression

(a)

A function \(f\) is called one-to-one if every element of the range corresponds to exactly one element of the domain.

That is, \(f\left( {{x_1}} \right) \ne f\left( {{x_2}} \right){\kern 1pt} {\kern 1pt} \) whenever \({\kern 1pt} {x_1} \ne {x_2}{\kern 1pt} \) or \(f\left( {{x_1}} \right) = f\left( {{x_2}} \right){\kern 1pt} {\kern 1pt} {\kern 1pt} \) whenever \({x_1} = {x_2}\).

Simplify the expression

(b)

Use horizontal line test to determine the graph is one-to-one.

A graph is one-to-one if and only if the graph of \(f\) intersects the horizontal line at most once.