In addition to linear, quadratic, rational, and also radical functions, there room A role of the type f(x) = bx, whereby b > 0 and also b ≠ 1.

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")">exponential functions
. Exponential features have the kind f(x) = bx, where b > 0 and b ≠ 1. Simply as in any exponential expression, b is called the The expression that is being raised to a power when using exponential notation. In 53, 5 is the base, i beg your pardon is the number the is continuously multiplied. 53 = 5 • 5 • 5. In ab, a is the base.

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and x is called the When a number is to express in the type ab, b is the exponent. The exponent shows how many times the base is supplied as a factor. Power and exponent median the exact same thing.

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.

An instance of one exponential duty is the growth of bacteria. Some bacteria twin every hour. If you start with 1 bacterium and it doubles every hour, friend will have 2x bacteria after ~ x hours. This have the right to be composed as f(x) = 2x.

Before girlfriend start, f(0) = 20 = 1

After 1 hour f(1) = 21 = 2

In 2 hours f(2) = 22 = 4

In 3 hrs f(3) = 23 = 8

and for this reason on.

With the an interpretation f(x) = bx and also the constraints that b > 0 and that b ≠ 1, the domain of one exponential function is the collection of all genuine numbers. The range is the set of all positive real numbers. The following graph reflects f(x) = 2x.

Exponential Growth

As you can see above, this exponential duty has a graph that gets an extremely close to the x-axis together the graph extends come the left (as x becomes an ext negative), but never yes, really touches the x-axis. Discovering the basic shape of the graphs of exponential functions is advantageous for graphing particular exponential equations or functions.

Making a table of values is likewise helpful, because you deserve to use the table to ar the curve the the graph more accurately. One thing to remember is the if a base has actually a an unfavorable exponent, then take the reciprocal of the basic to make the exponent positive. Because that example,

.

 Example Problem Make a table of worths for f(x) = 3x. x f(x)

Make a “T” to start the table through two columns. Label the columns x and also f(x).

x

 f(x) −2 −1 0 1 2

Choose number of values because that x and put them as separate rows in the x column.

Tip: that always good to include 0, optimistic values, and an adverse values, if girlfriend can.

x

 f(x) −2 −1 0 1 1 3 2 9

Evaluate the duty for each worth of x, and also write the result in the f(x) column alongside the x value you used. Because that example, when

x = −2, f(x) = 3-2 =

= , so  goes in the f(x) column beside −2 in the x column. F(1) = 31 = 3, for this reason 3 walk in the f(x) column alongside 1 in the x column.

Note that your table of values may be different from who else’s, if you decided different numbers for x.

Look in ~ the table of values. Think about what happens together the x values increase—so execute the function values (f(x) or y)!

Now that you have a table the values, you have the right to use these values to assist you draw both the shape and location that the function. Attach the point out as ideal you have the right to to do a smooth curve (not a series of straight lines). This mirrors that all of the clues on the curve are component of this function.

 Example Problem Graph f(x) = 3x. x f(x) −2 −1 0 1 1 3 2 9

Start with a table the values, like the one in the instance above.

x

 f(x) point −2 (−2, ) −1 (−1, ) 0 1 (0, 1) 1 3 (1, 3) 2 9 (2, 9)

If girlfriend think that f(x) together y, every row creates an notified pair that you have the right to plot top top a coordinate grid.

Plot the points.

Connect the point out as ideal you can, making use of a smooth curve (not a series of straight lines). Use the shape of an exponential graph to assist you: this graph gets an extremely close to the

x- axis top top the left, however never really touches the x-axis, and also gets steeper and steeper top top the right.

This is an instance of An exponential role of the type f(x) = bx, where b > 1, and b ≠ 1. The duty increases together x increases.

")">exponential growth
. Together x increases, f(x) “grows” an ext quickly. Let’s try another one.

 Example Problem Graph f(x) = 4x. x f(x) −2 −1 0 1 1 4 2 16

Start through a table of values. You can select different values, yet once again, it’s valuable to incorporate 0, some optimistic values, and also some an unfavorable values.

Remember,

4-2 =

= .

If friend think of f(x) as y, each row creates an notified pair the you deserve to plot top top a coordinate grid.

Plot the points.

Notice the the bigger base in this problem made the duty value skyrocket. Also with x as little as 2, the duty value is too large for the axis scale you supplied before. You can adjust the scale, however then our various other values are really close together. You might also try other points, such as as soon as x =

. Due to the fact that you recognize the square source of 4, friend can find that value in this case:
. The suggest
is the blue suggest on this graph.

For other bases, you could need to usage a calculator to help you uncover the role value.

Connect the points as finest you can, using a smooth curve (not a collection of straight lines). Usage the shape of an exponential graph to aid you: this graph gets really close to the x-axis top top the left, however never yes, really touches the

x- axis, and gets steeper and steeper ~ above the right.

Let’s compare the three graphs did you do it seen. The features f(x) = 2x, f(x) = 3x, and also

f(x) = 4x space all graphed below.

Notice that a larger base renders the graph steeper. A larger base likewise makes the graph closer to the y-axis for x > 0 and also closer to the x-axis because that x

Exponential Decay

Remember that for exponential functions, b > 0, but b ≠ 1. In the examples above, b > 1. What happens as soon as b is in between 0 and also 1, 0 b

 Example Problem Graph . x f(x) −2 4 −1 2 0 1 1 2

Start v a table of values.

Be careful with the an adverse exponents! psychic to take it the mutual of the base to do the exponent positive. In this case,

, and
.

Use the table together ordered pairs and also plot the points.

Since the points space not ~ above a line, girlfriend can’t usage a straightedge. Attach the clues as best you can using a smooth curve (not a collection of straight lines).

Notice the the shape is similar to the shape as soon as b > 1, yet this time the graph it s okay closer come the x-axis once x > 0, quite than once x one exponential duty of the form f(x) = bx, wherein 0 b . The function decreases together x increases.

")">exponential decay. Rather of the role values “growing” together x values increase, together they did before, the role values “decay” or decrease together x worths increase. They get closer and also closer to 0.

 Example Problem Graph . x f(x) −2 16 -1 4 0 1 1 2

Create a table of values. Again, be cautious with the negative exponents. Remember to take the mutual of the base to do the exponent positive.

.

Notice that in this table, the x worths increase. The y worths decrease.

Use the table bag to plot points. You may want to include new points, especially when one of the points indigenous the table, right here (−2, 16) i will not ~ fit on her graph. Due to the fact that you understand the square source of 4, try x =. Girlfriend can uncover that value in this case:

.

The point (, 8) has been consisted of in blue. You may feel it necessary to include extr points. You likewise may must use a calculator, depending on the base.

Connect the point out as best you can, making use of a smooth curve.

Which of the following is a graph for

?

A)

B)

C)

D)

A)

Incorrect. This graph is increasing, since the f(x) or y values boost as the x worths increase. (Compare the values for x = 1 and x = 2.) This graph shows exponential growth, through a base greater than 1. The exactly answer is Graph D.

B)

Incorrect. This graph is decreasing, however all the function values space negative. The variety for an exponential role is always positive values. The exactly answer is Graph D.

C)

Incorrect. This graph is increasing, yet all the duty values space negative. The correct graph should be decreasing with positive role values. The exactly answer is Graph D.

D)

Correct. All the duty values space positive, and also the graph is decreasing (showing exponential decay).

Applying Exponential Functions

Exponential attributes can be offered in plenty of contexts, such as compound attention (money), population growth, and radioactive decay. In many of these, however, the function is not exactly of the type f(x) = bx. Often, this is readjusted by including or multiply constants.

For example, the compound interest formula is

, wherein P is the principal (the initial invest that is gathering interest) and A is the amount of money you would certainly have, through interest, in ~ the finish of t years, making use of an yearly interest price of r (expressed together a decimal) and also m compounding durations per year. In this case the base is the value represented by the expression 1 +
and the exponent is mt—a product of 2 values.

 Example Problem If you invest \$1,000 in an account paying 4% interest, compounded quarterly, exactly how much money will you have actually after 3 years? The money you will have actually after 3 years will certainly be A. P = \$1,000 r = 0.04 m = 4 t = 3 First determine which the A, P, r, m, and t is being asked for, then recognize values because that the staying variables. The major is \$1,000. The price is 4% = 0.04. The time in year is 3. Compounded quarterly means 4 time a year. To uncover the amount A, usage the formula. Answer You will have actually \$1,126.83 ~ 3 years. Round the number to the nearest cent (hundredth). Notice the this way the lot of attention earned after 3 years is \$126.83. (\$1,126.83, minus the principal, \$1,000).

Radioactive decay is an example of exponential decay. Radioactive elements have a half-life. This is the amount of time the takes for half of a mass of the aspect to degeneration into one more substance. Because that example, uranium-238 is a gradually decaying radioactive element with a half-life of around 4.47 billion years. That means it will take that lengthy for 100 grams of uranium-238 come turn right into 50 grams of uranium-238 (the various other 50 grams will have turned into an additional element). It is a lengthy time! on the various other extreme, radon-220 has actually a half-life of about 56 seconds. What walk this mean? 100 grams the radon-220 will certainly turn into 50 grams that radon-220 and also 50 grams the something rather in less than a minute!

Since the quantity is halved each half-life, one exponential duty can be supplied to describe the amount remaining over time. The formula

gives the remaining amount R native an initial lot A, where h is the half-life that the element and also t is the lot of time happen (using the same time unit together the half-life).

 Example Problem Caesium-137 is a radioactive element used in clinical applications. It has actually a half-life of around 30 years. Expect a laboratory has 10 grams the caesium-137. If lock don’t usage it, just how much will still be caesium-137 in 60 years? R: This is the continuing to be value, what you are trying to find. A: The initial amount was 10 grams. h: The half-life is 30 years. t: The lot of time happen is 60 years. (Note that this is in the exact same unit, years, together the half-life.) Identify the values known in the formula. Use the formula. Answer There will certainly be 2.5 grams that caesium-137 in 60 years.