Question

How do you identify equations as exponential growth, exponential decay, linear growth or linear decay f(x) = $4{{\left( \dfrac{3}{2} \right)}^{x}}$?

Answer 1

In this problem, we have to identify that the given function is showing exponential growth or exponential decay. Apart from that we have to know if this function possesses linear growth or linear decay. Exponential growth and decay are identified by positive or negative values on which exponent is given. And linear identifies that graph is a curve or a straight line.

Complete step by step answer:
Now, let’s begin with the identification.
As we know that an exponential function is that mathematical function which is in the form of f(x) = ${{a}^{x}}$, where ‘x’ is a variable and ‘a’ is a constant. And ‘a’ is the base of the function and it should be greater than 0 while ‘x’ is the power or exponent. When we talk about exponential growth, it means if the value is positive on which exponent is given, then that function is growing i.e. x is increasing. But on the other hand, decay means the value is decreasing and it is only possible if the value on which the exponent is given, is negative. Linear term refers to the straight line. A function can grow or decay linearly too.
In this particular question, the function is showing exponential growth because we are given ‘x’ as an exponent on the value $\dfrac{3}{2}$ which is a positive value. So the function is growing as the power will increase.

Note: Students should not get confused in the terms exponential growth and linear growth. The function having linear growth or decay does not contain any exponent. Exponents will always give you a curve rather than a straight line.