as x approaches ∞, which statement is correct? a the linear function will eventually exceed the exponential…

as x approaches ∞, which statement is correct? a the linear function will eventually exceed the exponential function. b the exponential function will eventually exceed the linear function. c the linear function will only exceed the exponential function if its initial value is greater. d the exponential function will only exceed the linear function if its initial value is greater.

as x approaches ∞, which statement is correct? a the linear function will eventually exceed the exponential function. b the exponential function will eventually exceed the linear function. c the linear function will only exceed the exponential function if its initial value is greater. d the exponential function will only exceed the linear function if its initial value is greater.

Answer

Explanation:

Step1: Recall function - growth properties

Exponential functions ($y = a\cdot b^x$, $b>1$) grow faster than linear functions ($y = mx + c$) as $x\rightarrow\infty$.

Step2: Analyze each option

  • Option A: Incorrect as exponential grows faster.
  • Option B: Correct. As $x\rightarrow\infty$, exponential function exceeds linear.
  • Option C: Incorrect, growth - rate is key, not initial value.
  • Option D: Incorrect, growth - rate is key, not initial value.

Answer:

B. The exponential function will eventually exceed the linear function.