NZ Level 7 (NZC) Level 2 (NCEA)
Characteristics and Domain and Range of Exponential Functions
Lesson

## Why study the exponential function?

The exponential function is used in real life to model certain types of growth and decay. Let's think about a  simple example to explore the idea.

Consider a house, valued at $\$50000$$50000 that is set to increase in value by 1010% each year. In the first year, the house's value will become \550000$$550000 (determined by the calculation $500000+10%\left(500000\right)=550000$500000+10%(500000)=550000). In the second year, the house's value becomes $\$605000$$605000, which is an increase of \55000$$55000 over the previous year.

So even though the house increased by the same $10$10% each year, the actual amount of increase was more in the second year than that of the first year. This is a key idea. The rate of growth in dollars in the house price is proportional to the house price itself. Less expensive houses will grow in value much slower than expensive houses.

Alternatively If an item depreciates (loses value) by $10$10% each year, then the largest drop in value will occur in the first year. As the years go on, the amount of fall in dollar value will accordingly reduce. It will never reduce to zero because we are always calculating $10$10% of something.

The basic mathematical model for this type of process has the independent variable in the exponent. Typically, the model has the general form $y=Ab^{mx+c}+k$y=Abmx+c+k, The number $b$b is called the base of the function and the other constants ($A$A, $m$m $c$c, and $k$k) are there to allow the model to match the particular real life growth or decay rate under consideration.

For example, without giving the details of the calculation, the house price example can be modelled by the function $y=500000\times10^{0.041393x}$y=500000×100.041393x where $x$x is the number of years of growth (note that $A=500000$A=500000, the base $b=10$b=10 and the growth rate $m$m is the fraction $0.41393$0.41393). The point being that we can use the function to determine future values of the house. This is the power of models.

## Domain and Range

Like any function, we are interested in knowing the domain and range of the exponential function.

Remember that the domain is the set of values that the independent variable (usually $x$x) can take, and the range is the set of values that the dependent variable (usually $y$y) can take.

Unless there are restrictions imposed on the function, the natural domain of the general exponential function $y=Ab^{mx+c}+k$y=Abmx+c+k is the entire set of real numbers. So for functions like $y=2^x$y=2x$y=3^{-x}$y=3x$y=\left(0.5\right)^x+3$y=(0.5)x+3$y=5-10^x$y=510x etc all have domains given by $x\in\Re$x

This means that any real value of $x$x can be substituted into any exponential function of this form to calculate  an associated $y$y value.  For example, at $x=0$x=0, the function $y=2^x$y=2x becomes $2^0=1$20=1, and so the y-intercept of the function $y=2^x$y=2x is $1$1.

The $y$y-intercept of the general function $y=Ab^{mx+c}+k$y=Abmx+c+k is determined by setting $x=0$x=0. So for $y=\left(0.5\right)^x+3$y=(0.5)x+3, the $y$y intercept is $y=\left(0.5\right)^0+3=1+3=4$y=(0.5)0+3=1+3=4

C

### Outcomes

#### M7-2

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

#### 91257

Apply graphical methods in solving problems