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The exponential function f is graphed in the xy-plane. As x increases by 1,y increases by a factor of 3 . Which of the following could be f ? Choose 1 answer: (A) f(x)=((1)/(3))^(x) (B) f(x)=((1)/(3))^(x)+3 (C) f(x)=3^(x)+2 (D) f(x)=2(3)^(x)

The exponential function f f is graphed in the xy x y -plane. As x x increases by 1,y 1, y increases by a factor of 33 . Which of the following could be f f ?\newlineChoose 11 answer:\newline(A) f(x)=(13)x f(x)=\left(\frac{1}{3}\right)^{x} \newline(B) f(x)=(13)x+3 f(x)=\left(\frac{1}{3}\right)^{x}+3 \newline(C) f(x)=3x+2 f(x)=3^{x}+2 \newline(D) f(x)=2(3)x f(x)=2(3)^{x}

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Q. The exponential function f f is graphed in the xy x y -plane. As x x increases by 1,y 1, y increases by a factor of 33 . Which of the following could be f f ?\newlineChoose 11 answer:\newline(A) f(x)=(13)x f(x)=\left(\frac{1}{3}\right)^{x} \newline(B) f(x)=(13)x+3 f(x)=\left(\frac{1}{3}\right)^{x}+3 \newline(C) f(x)=3x+2 f(x)=3^{x}+2 \newline(D) f(x)=2(3)x f(x)=2(3)^{x}
  1. Identify Base of Exponential Function: We are given that as xx increases by 11, yy increases by a factor of 33. This means that the base of the exponential function must be 33. We can eliminate any options that do not have a base of 33.
  2. Eliminate Options with Incorrect Bases: Option (A) f(x)=(13)xf(x) = (\frac{1}{3})^x has a base of 13\frac{1}{3}, which would mean that yy decreases as xx increases, because 13\frac{1}{3} is less than 11. This does not match the description given in the problem.
  3. Analyze Option (A): Option (B) f(x)=(13)x+3f(x) = (\frac{1}{3})^x + 3 also has a base of 13\frac{1}{3}, which, as previously stated, would mean that yy decreases as xx increases. The addition of 33 does not change the fact that the base is still 13\frac{1}{3}, so this option is also incorrect.
  4. Analyze Option (B): Option (C) f(x)=3x+2f(x) = 3^x + 2 has a base of 33, which fits the description of the problem. However, we need to check if the ＂+22＂ affects the property that yy increases by a factor of 33 as xx increases by 11.
  5. Analyze Option (C): The ＂+22＂ in option (C) is a vertical shift of the graph of the function f(x)=3xf(x) = 3^x. It does not affect the factor by which yy increases when xx increases by 11. The factor of increase is still determined by the base of the exponential, which is 33. Therefore, option (C) could represent the function described in the problem.
  6. Analyze Option (D): Option (D) f(x)=2(3)xf(x) = 2(3)^x has a base of 33, which is correct. However, the factor of 22 in front of the base means that yy is not just increasing by a factor of 33 as xx increases by 11, but rather by a factor of 2×32 \times 3. This does not match the description given in the problem.

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