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Problem-Solving and Data Analysis / Two-variable data: Models and scatterplots Difficulty: Medium

Which of the following is true about the values of $2 to the x power$ and $2 x plus 2$ for $x greater than 0$ ?

For all $x greater than 0$ , it is true that $2 to the x power, is less than 2 x plus 2$ .

For all $x greater than 0$ , it is true that $2 to the x power, is greater than 2 x plus 2$ .

There is a constant c such that if $0 is less than x, which is less than c$ , then $2 to the x power, is less than 2 x plus 2$ , but if $x is greater than c$ , then $2 to the x power, is greater than 2 x plus 2$ .

There is a constant c such that if $0 is less than x, which is less than c$ , then $2 to the x power, is greater than 2 x plus 2$ , but if $x is greater than c$ , then $2 to the x power, is less than 2 x plus 2$ .

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Explanation

Choice C is correct. At $x equals 0$ , the value of $2 raised to the x power$ is less than the value of $2 x plus 2, where 2 to the 0 power is less than, 2 times 0, plus 2$ , which is equivalent to $1 is less than 2$ . As the value of x increases, the value of $2 raised to the x power$ remains less than the value of $2 x plus 2$ until $x equals 3$ , which is when the two values are equal: $2 cubed equals, 2 times 3, plus 2$ , which is equivalent to $8 is equal to 8$ . Then, for $x greater than 3$ , the value of $2 raised to the x power$ is greater than the value of $2 x plus 2$ . So there is a constant, 3, such that when $0 is less than x, which is less than 3$ , then $2 raised to the x power is less than, 2 x plus 2$ , but when $x is greater than 3$ , then $2 raised to the x power is greater than, 2 x plus 2$ .

Choice A is incorrect because $2 raised to the x power is greater than, 2 x plus 2$ when $x is greater than 3$ . Choice B is incorrect because $2 raised to the x power is less than, 2 x plus 2$ when $0 is less than x, which is less than 3$ . Choice D is incorrect because $2 raised to the x power is less than, 2 x plus 2$ when $0 is less than x, which is less than 3$ and $2 raised to the x power is greater than, 2 x plus 2$ when $x is greater than 3$ .