Q: Consider the function $$f(x) = |x|. Is $$this function one-to-one? Explain.

No, because $$f(-a) = f(a)$$ for $$a \neq 0$$, so a horizontal line at $$y = a$$ for $$a \u003e 0$$ intersects the graph at two points.

Question 1

Given the graph of the function $$f(x) = -x^2$ + 4x - 3$$, determine whether the function is one-to-one. Justify your answer using the horizontal line test.

Accepted Answer

No, there is a horizontal line that intersects the graph at more than one point.

Question 2

Which of the following statements best describes the horizontal line test for one-to-one functions?

Accepted Answer

A function is one-to-one if every horizontal line intersects its graph at exactly one point.

Question 3

Suppose a function $$f(x)$$ passes the vertical line test but fails the horizontal line test. Which of the following is true?

Accepted Answer

The function is a function but not one-to-one.

Question 4

If a function $$f(x) is $$not one-to-one, which of the following must be true?

Accepted Answer

There exists at least one horizontal line that intersects the graph of $$f(x) at $$more than one point.

Question 5

Which of the following real-world scenarios can be modeled by a one-to-one function?

Accepted Answer

Assigning each student in a class a unique student ID number.

Question 6

Given the function $$f(x) = x^3$ - 2x + 1$$, is this function one-to-one? Why or why not?

Accepted Answer

Yes, because every horizontal line intersects the graph at exactly one point.

Question 7

A hospital records the body temperature of patients every hour. If the temperature at 2 PM and 4 PM is the same for a patient, what does this say about the function mapping time to temperature?

Accepted Answer

The function is not one-to-one because two different times have the same temperature.

Question 8

Which of the following is a necessary condition for a function $$f(x) to $$have an inverse that is also a function?

Accepted Answer

The function $$f(x)$$ must be one-to-one.

Question 9

Consider the function $$f(x) = |x|. Is $$this function one-to-one? Explain.

Accepted Answer

No, because $$f(-a) = f(a)$$ for $$a 
eq 0$$, so a horizontal line at $$y = a$$ for $$a \u003e 0$$ intersects the graph at two points.

Question 10

A chemical reaction's rate $$R is $$given as a function of temperature $$T by$$ $$R(T) = 2T - 5. Is $$this function one-to-one? What does this mean in the context of the reaction?

Accepted Answer

Yes, because for every rate $$R$$, there is exactly one temperature $$T$$ that produces it.

Analyzing One-to-One Functions Using Graphs

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