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A function from a set to a set is a relation that assigns to each element in the set exactly one element in the set . The set is the domain (also called the set of inputs) of the function and the set contains the range (also called the set of outputs). .1. It isn t an exponential function.We have the following equation:That can be written as:Recall that the definition of exponential functions establishes that:.That is:In this problem, , therefore this is not an exponential function.2. Growth.The...

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CatelynBGray

24.10.2021

Mathematics

A function from a set to a set is a relation that assigns to each element in the set exactly one element in the set . The set is the domain (also called the set of inputs) of the function and the set contains the range (also called the set of outputs).

$We \ denote \ the \ \mathbf{exponential \ function} \ f \ with \ base \ a \ as: \\ \\ f(x)=a^x \\ \\ where \ a0, \ a\neq 1, \ and \ x \ is \ any \ real \ number$ .

1. It isn't an exponential function.

We have the following equation:

y=100(1-12)^t

That can be written as:

y=100(-11)^t

Recall that the definition of exponential functions establishes that:

$We \ denote \ the \ \mathbf{exponential \ function} \ f \ with \ base \ a \ as: \\ \\ f(x)=a^x \\ \\ where \ a0, \ a\neq 1, \ and \ x \ is \ any \ real \ number$ .

That is:

$a \ \mathbf{must} \ be \ greater \ than \ 1$

In this problem, a=-11 , therefore this is not an exponential function.

2. Growth.

The function:

y=0.1(1.25)^t

is an exponential function because is a function of the form $f(t)=ka^t \\ \\ where \ a0 \ and \ k \ constant$

So $k=0.1 \ and \ a=1.25$ . Since $a \ is \ greater \ than \ 1$ and being raised to the power of , the function increases. This means that increases as increases as illustrated in Figure 1. This represents a growth.

3. Growth.

The function:

$y=((1-0.03)12)^{2t}$ can be written as:

$y=11.64^{2t}$

and is an exponential function because is a function of the form $f(t)=a^{bt} \\ \\ where \ a0 \ and \ b \ constant$

So $a=11.64 \ and \ b=2$ . Since $a \ is \ greater \ than \ 1$ and being raised to the power of , the function increases. As in the previous exercise, this means that increases as increases as illustrated in Figure 2. This represents a growth.

4. Decay.

The function:

y=426(0.98)^t

is an exponential function because is a function of the form $f(t)=ka^t \\ \\ where \ a0 \ and \ k \ constant$

So $k=426 \ and \ a=0.98$ . Since $a \ is \ less \ than \ 1$ and being raised to the power of , the function decreases. Here this means that decreases as increases as illustrated in Figure 3. This represents a decay.

5. Growth.

The function:

y=2050(12)^t

is an exponential function because is a function of the form $f(t)=ka^t \\ \\ where \ a0 \ and \ k \ constant$

So $k=2050 \ and \ a=12$ . Since $a \ is \ greater \ than \ 1$ and being raised to the power of , the function increases. So in this function also increases as increases as illustrated in Figure 4. This represents a growth.

Does each function describe exponential growth or decay? drag and drop the equations into the boxes

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