Question

Find the range of ƒ(x) = 3 · 2^x + 10 + 9. 

Answer 1

The general form of an exponential function is defined by f (x ) = c*a ^x-h + k

where a  is positive constant , a  not equals to 1 and a is called base. x is any real number.

The graph has horizontal asympotote of y = k . and passes through (h , c +k )

f (x ) = 3 * 2^x + 10 + 9

f (x ) = 3 *  2^x + 19

Horizontal asympotote y = 19 and passes through (0 , 22)

y  = 3 * 2^x + 19

We will find the range by looking at the graph of the exponential function.

Choose values for y and find the corresponding values for x.

x	y  = 3 * 2x + 19	(x, y )
-2	y = 3*2-2 + 19 = 3/4 +19 = 19.75	(-2,19.75)
-1	y = 3*2-1 + 19 = 3/2 + 19 = 20.5	(-1, 20.5)
0	y  = 3*20+ 19 = 3 + 19 = 22	(0, 22)
1	y = 3*21 + 19 = 6 +19 = 25	(1, 25)

Graph

1.Draw a coordinate plane.

2.Plot the coordinate points and dash the horizontal asympotote.

3.Then sketch the graph, connecting the points with a smooth curve.

We know that range is all possible y  values.

The range is all positive real numbers (never zero).

Range .

 

Answer 2

In general, an equation of the form , where a ≠ 0, b > 0, and b ≠ 1, is called an exponential function with base b.

The standard form of an exponential function is .

For , the function is defined for all real values of x. Therefore, the domain is .

The range of is dependent on the sign of a.

To find the range of , start with , since the definition of exponential function.

The function is f (x ) = 3.2 ^x + 10 + 9 ⇒ f (x ) = 3.2 ^x + 19.

Compare the equation f (x ) = 3.2 ^x + 19 with standard form of an exponential function is .

y = f (x ), a = 3 > 0, b = 2 and q = 19.

To find the range of f (x ) = 3.2 ^x + 19 , start with .

Multiply each side by 3 to the inequality.

3.2 ^x > 3.0

3.2 ^x > 0

Add 19 to each side of the inequality.

3.2 ^x + 19 > 0 + 19

f (x ) > 19.

The range is { f (x ) : f (x ) ∈ R, f (x ) > 19 }.