Subject Test Math 2

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Subject Test Math 2

Question 15 of 28

What is the range of the function defined by $f (x) = \frac{1}{x} + 2$ ?

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All real numbers

Correct Answer:

No

All real numbers except $negative 1 half$

Correct Answer:

No

All real numbers except 0

Correct Answer:

No

All real numbers except 2

Correct Answer:

Yes

All real numbers between 2 and 3

Correct Answer:

No

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The correct answer is D.

The range of the function defined by $f (x) = \frac{1}{x} + 2$ is the set of $y - values$ such that $y = \frac{1}{x} + 2$ for some $x - value$ .

One way to determine the range of the function defined by $f (x) = \frac{1}{x} + 2$ is to solve the equation $y = \frac{1}{x} + 2$ for $x$ and then determine which $y - values$ correspond to at least one $x - value$ . To solve $y = \frac{1}{x} + 2$ for $x$ , first subtract $2$ from both sides to get $y - 2 = \frac{1}{x}$ and then take the reciprocal of both sides to get $x = \frac{1}{y - 2}$ . The equation $x = \frac{1}{y - 2}$ shows that for any $y - value$ other than $2$ , there is an $x - value$ such that $y = \frac{1}{x} + 2$ , and that there is no such $x - value$ for $y = 2$ . Therefore, the range of the function defined by $f (x) = \frac{1}{x} + 2$ is all real numbers except $2$ .

Alternatively, one can reason about the possible values of the term $\frac{1}{x}$ . The expression $\frac{1}{x}$ can take on any value except $0$ , so the expression $\frac{1}{x} + 2$ can take on any value except $2$ . Therefore, the range of the function defined by $f (x) = \frac{1}{x} + 2$ is all real numbers except $2$ .

Question Difficulty:

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Subject Test Math 2

Question 15 of 28

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