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Both of these functions grow as x x gets larger and larger. Which function eventually exceeds the other?\newlineChoices:\newline(A) f(x)=2x+9(A) \ f(x) = 2x + 9 \newline(B) g(x)=2x(B)\ g(x) = 2^x

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Q. Both of these functions grow as x x gets larger and larger. Which function eventually exceeds the other?\newlineChoices:\newline(A) f(x)=2x+9(A) \ f(x) = 2x + 9 \newline(B) g(x)=2x(B)\ g(x) = 2^x
  1. Identify function types: Identify the types of functions for f(x)f(x) and g(x)g(x).\newlinef(x)=2x+9f(x) = 2x + 9 is a linear function because it is of the form y=mx+by = mx + b.\newlineg(x)=2xg(x) = 2^x is an exponential function because the variable xx is in the exponent.
  2. Compare growth rates: Compare the growth rates of linear and exponential functions.\newlineExponential functions grow at a faster rate than linear functions as xx becomes very large.
  3. Determine exceeding function: Determine which function will eventually exceed the other.\newlineSince g(x)=2xg(x) = 2^x is an exponential function and f(x)=2x+9f(x) = 2x + 9 is a linear function, and we know that exponential functions grow faster than linear functions, g(x)=2xg(x) = 2^x will eventually exceed f(x)=2x+9f(x) = 2x + 9 as xx gets larger.

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