Question

Given the formula for an exponential function, it is possible to determine whether the function grows or decays exponentially just by looking at the formula? Explain why or why not and give an example in your explanation.

Answer 1

An exponential function is one in which the independent variable x appears in the exponent and has a constant a as its base.

`f(x)=a^x`

To know if the function is increasing or decreasing we have to take into account whether the sign of the constant a

For it to be increasing, the constant a must be greater than 1

`a>1`

In the image we can see 2 increasing exponential functions in blue and green where the blue constant is 5 and the green constant is 2 both constants are greater than 1

For an exponential function to be decreasing, the value of the constant a must be between 0 and 1

[tex]0In the image we can see 2 decreasing exponential functions in red and purple where the purple constant is 0.4 and the red constant is 0.6 both constants are between 0 and 1