Question

Which of the following functions increases the fastest as x values get larger? of (x) = 5x – 3 Of(x) = 53 f (x) = 3! + 20 f(x) = 3x + 20

Answer 1

Rate of change

Exponential functions increase faster than linear functions because the variable is at the exponent.

We have two exponential functions and two linear functions.

The exponential functions are

`f\mleft(x\mright)=5^x-3`

And

`f\mleft(x\mright)=3^x+20`

We can compare which of the exponential function grow faster by substituting x for two values.

Select x=1 and x=2

The first equation for both values is:

`\begin{gathered} f(1)=5^1-3=2 \\ f(2)=5^2-3=23 \\ f(2)-f(1)=23-2=21 \end{gathered}`

Now for the second function:

`\begin{gathered} f(1)=3^1+20=23 \\ f(2)=3^2+20=29 \\ f(2)-f(1)=29-23=6 \end{gathered}`

Therefore, the function that increases the fastest is

`f(x)=5^x-3`

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