Question

Matilda has been studying exponential functions in her algebra class and has started to notice a pattern. For some exponential functions, when she plugs in larger and larger numbers, she gets larger and larger numbers. For others, she gets smaller and smaller numbers. And for some, plugging in larger and larger numbers actually gets her closer and closer to one number. Which of the following exponential functions get closer and closer to one value as you plug-in larger and larger values for x?

Answer 1

Answer:

• Some examples of functions for which plugging in larger and larger numbers actually gets her closer and closer to one number are:

                  $y=3(1/x)^5\\\\y=1,500x^{-1}\\\\y=400+1,500x^{-1}$

Explanation:

The exponential functions that get larger and larger numbers when you plug in larger and larger numbers are called growing exponential functions and they are those with an exponential rate greater than 1.

              $y=A\cdot B^x,\text{ where }A\neq 0,B>0$

The exponential functions that get smaller and smaller numbers when you plug in larger and larger numbers are called decaying exponential functions and they are those with a positive exponential rate less than 1.

               $y=A\cdot B^x,\text{ where }A\neq 0,0$

The exponential functions that  plugging in larger and larger numbers actually gets her closer and closer to one number are those with a negative exponential rate less.

               $y=A\cdot B^x,\text{ where }A\neq 0,B$

Some examples of this kind of equations are:

• $y=3x^{-5}$,  or its equivalent form $y=3(1/x)^5$

When x gets larger and larger, $(1/x)^5$ gets closer and closer to , and the function gets closer and closer to 0.

If you add a constant to the parent function then the function will get closer and closer to the value of the constant.

For instance, for the function   $y=3(1/x)^5+20$, plugging in  larger and larger numbers actually gets her closer and closer to 20.