Question

Which graph represents functions f and g? f: Initial value of 200 decreasing at a rate of 4%. g: Initial value of 40 increasing at a rate of 8%. The graphs are labeled A, B, C, D. Tell which graph represents the function f and g.

Answer 1

We are given two exponential functions- Function "f" has an initial value of 200 and decreases at a rate of 4%. This means that "f" is a function of the form:

`f(x)=a_0(1-b)^x`

Where:

`\begin{gathered} a_0=\text{ initial value} \\ b=\text{ rate of decay in decimal form. } \end{gathered}`

To determine the rate of decay in decimal form we divide the percentage by 100:

`b=(4)/(100)=0.04`

Now, we substitute the values:

`f(x)=200(1-0.04)^x`

Solving the operation:

`f(x)=200(0.96)^x`

Since the function "f" decreases this means that its graph must have the following shape:

Where the initial value of 200 is the y-intercept. This means that the possible right options are B or C.

To determine which of the two options is the right option we need to also have into account that the function "g" is increasing, therefore, its graph must have the following shape:

Also, we will determine one point on the graph that is consistent with the function we determined for "f".

In graph B we notice the following point for "f":

`f(25)=25`

If B is the right option then when we substitute 25 in "f" we should get 25.

`f(25)=200(0.96)^(25)`

Solving the operations:

`f(25)=72.1`

Since we did not get 25 this means that B is not the graph.

Now, let's take a point in C:

`f(35)\approx50`

Now, let's substitute the value in the function:

`f(35)=200(0.98)^(35)`

Solving the operations:

`f(35)=47.9`

Since we get a number close to 50 this means that the right graph is C.