Linear Or Not?!

Question

Determine whether the following function is linear

 

y = 3^x + 5

 

Explain why

Answer 1

y = 3^x + 5 is a geometric growth function (for increasing positive values of x). It is nonlinear, as pointed out in the first answer, because all linear functions are represented by the first power of x. As an example of a linear function, consider the following:

 

y = 5x + 3

 

This is a linear function because the power of x is 1. Recall that x¹ = x.

 

Your original function is not even a polynomial function (e.g., x², x³, x⁴, and so on) because the relationship raises a constant to the power of x, rather than raising x to an integer power (as is the case with all polynomial functions).

 

Your original function is related to exponential functions by its form but not by its base. Common exponential functions are the inverse of Napierian or common logarithmic functions. Some examples of exponential functions are the following:

 

y = e^x     or     y = 10^x

 

These functions represent the inverse of the natural and common logarithms, ln(x) and log(x). The "e" in the exponential function is a number that is referred to as "Euler's number" or "Napier's constant," with a value of approximately 2.71828. It is a constant like pi (≈3.14159).

 

Thus y = 3^x +5 is a geometric function, not a linear function.

Answer 2

It is not linear because it is not of the form

 

f(x) = ax¹ + bx⁰ = ax + b.