The exponential function

Exponential functions and logarithms: Exponential functions

The exponential function

We call a function of the form \[f(x)=\blue{a}^x\] with #\blue{a}>0# an exponential function.

The graph is asymptotic to the #x#-axis. No matter how far left you go, #\blue{a}^x# will never be equal to #0.#

Dependent on the value of #\blue{a}#, the graph is increasing or decreasing. If #\blue{a}>1#, the graph is increasing. If #0<\blue{a}<1#, the graph is decreasing.

We note that the the entire #x#-axis is domain of the exponential function, and the range just the positive #y#-axis.

geogebra plaatje

Chess board
For numbers greater than #1#, the exponential function is a function which increases rapidly. There is a nice legend which illustrates this well: Sometime around the year 500, chess was invented in India. The king wanted to thank the inventor of the game, and promised the man a reward of his own choosing.

The inventor had the following idea: for the first square he wanted #1# rice grain, for the second square he asked for #2#, and for the third square asked for #4#, etcetera. The king was almost insulted, because he thought the man asked for too small of a reward.

However, when the members of the king's court started to calculate how many rice grains the man would receive, the king found that the reward was pretty large indeed! The total amount of rice grains, which is

\[2^0+2^1+2^2+\ldots+2^{62}+2^{63}=18446744073709551615,\]

happened to be more than the global production of rice.

a=1
An exception to the properties of the exponential function described above is #\blue{a}=1#. This graph is neither decreasing nor increasing. The range is also not the entire #y#-axis, but only #y=1#.

Rules
When calculating with exponential functions, we often use the rules described in Calculating with integer exponents, so make sure to review these. For example: \[\blue{2}^\orange{x}\cdot\blue{2}^\purple{x+4}=\blue{2}^{\orange{x}+\purple{x+4}}=\blue{2}^{2x+4}.\]

Indicate which of the following is an exponential function.

#f(x)=6^x#

#f(x)=x^3#

#f(x)=\left(\frac{1}{3}\right)^x#

#f(x)=\sqrt{x}#