Question

first up, lets review the assignments learning objectives
question
which of the following graphs show a function which is one - to - one?
select all correct answers
select all that apply

Explanation:

Step1: Recall the definition of one - to - one function

A function \(y = f(x)\) is one - to - one if for any \(x_1
eq x_2\) in the domain of \(f\), \(f(x_1)
eq f(x_2)\). Geometrically, a function is one - to - one if it passes the horizontal line test. That is, no horizontal line intersects the graph of the function more than once.

Step2: Analyze the first graph

The first graph (the non - linear, wavy graph) fails the horizontal line test. There exist horizontal lines that intersect the graph at more than one point.

Step3: Analyze the second graph

The second graph (the straight line with a non - zero slope) passes the horizontal line test. For a linear function \(y=mx + b\) where \(m
eq0\), if \(x_1
eq x_2\), then \(y_1=mx_1 + b\) and \(y_2=mx_2 + b\), and \(y_1 - y_2=m(x_1 - x_2)
eq0\) (since \(m
eq0\) and \(x_1
eq x_2\)).

Step4: Analyze the third graph

The third graph (the hyperbola - like graph in the first quadrant) passes the horizontal line test. For any two distinct \(x\) values in its domain, the \(y\) values are distinct.

Answer:

The second and the third graphs.