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.
")">base 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.
")">exponent.
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 xaxis together the graph extends come the left (as x becomes an ext negative), but never yes, really touches the xaxis. 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.
Answer
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) = 32 =
= , 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.
Answer
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 xaxis, 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,
42 =
= .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.
Answer
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 xaxis 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 yaxis for x > 0 and also closer to the xaxis 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.
Answer
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 xaxis 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.
Answer
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) Show/Hide Answer 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.
Radioactive decay is an example of exponential decay. Radioactive elements have a halflife. 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, uranium238 is a gradually decaying radioactive element with a halflife of around 4.47 billion years. That means it will take that lengthy for 100 grams of uranium238 come turn right into 50 grams of uranium238 (the various other 50 grams will have turned into an additional element). It is a lengthy time! on the various other extreme, radon220 has actually a halflife of about 56 seconds. What walk this mean? 100 grams the radon220 will certainly turn into 50 grams that radon220 and also 50 grams the something rather in less than a minute! Since the quantity is halved each halflife, 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 halflife that the element and also t is the lot of time happen (using the same time unit together the halflife).
