Analyze the form of the function, and compare it to the function from the previous exercise to understand why they do or do not have the same characteristics.
B
b
Analyze the form of the function, and compare it to the function from the previous exercise to understand why they do or do not have the same characteristics.
C
c
Analyze the form of the function, and compare it to the function from the previous exercise to understand why they do or do not have the same characteristics.
D
d
Analyze the form of the function, and compare it to the function from the previous exercise to understand why they do or do not have the same characteristics.
E
e
Analyze the form of the function, and compare it to the function from the previous exercise to understand why they do or do not have the same characteristics.
F
f
Analyze the form of the function, and compare it to the function from the previous exercise to understand why they do or do not have the same characteristics.
A
a
See solution.
B
b
See solution.
C
c
See solution.
D
d
See solution.
E
e
See solution.
F
f
See solution.
Practice makes perfect
a
Let's start by graphing the function f(x) = 2^x, so that we can check if it has the same characteristics that we identified in the previous exercise and why.
Now that we have the graph at hand, let's see which characteristics this function shows and analyze the function's form to understand why.
As x takes bigger negative values, the function approaches 0. ✓
Reason: We can use the negative exponent definition to write f(x) = 2^x as f(x) = 12^x for negative exponent values. Note than the bigger the negative value x takes, the smaller the value the function takes.
The domain of the function is all real numbers. ✓
Reason: A power with a positive base is defined for any exponent value.
The range of the function is all real numbers greater than zero. ✓
Reason: A power with a positive base cannot take negative values nor be 0.
When x=0, the value of the function equals the coefficient multiplying the power. In this case, f(0) = 1. ✓
Reason: For any nonzero number a, a^0=1. For this case, f(0) = 2^0 = 1.
b
The graph of the function f(x) = 2(3)^x is shown below.
Now that we have the graph at hand, let's see which characteristics this function shows and analyze the function's form to understand why.
As x takes bigger negative values, the function approaches 0. ✓
Reason: We can use the negative exponent definition to write f(x) = 2(3)^x as f(x) = 23^x for negative exponent values. Note than the bigger the negative value x takes, the smaller the value the function takes.
The domain of the function is all real numbers. ✓
Reason: A power with a positive base is defined for any exponent value.
The range of the function is all real numbers greater than zero. ✓
Reason: A power with a positive base cannot take negative values, nor be 0.
When x=0, the value of the function equals the coefficient multiplying the power. In this case, f(0) = 2. ✓
Reason: For any nonzero number a, a^0=1. For this case, f(0) = 2(3^0) =2(1)=2.
c
The graph of the function f(x) = 3(1.5)^x is shown below.
Now that we have the graph at hand, let's see which characteristics this function shows and analyze the function's form to understand why.
As x takes bigger negative values, the function approaches 0. ✓
Reason: We can use the negative exponent definition to write f(x) = 3(1.5)^x as f(x) = 31.5^x for negative exponent values. Note than the bigger the negative value x takes, the smaller the value the function takes.
The domain of the function is all real numbers. ✓
Reason: A power with a positive base is defined for any exponent value.
The range of the function is all real numbers greater than zero. ✓
Reason: A power with a positive base cannot take negative values, nor be 0.
When x=0, the value of the function equals the coefficient multiplying the power. In this case, f(0) = 3. ✓
Reason: For any nonzero number a, a^0=1. For this case, f(0) = 3(1.5^0) = 3(1)=3.
d
The graph of the function f(x) = ( 12)^x is shown below.
Now that we have the graph at hand, let's check which characteristics this function shows and analyze the function's form to understand why.
As x takes bigger negative values, the function approaches 0. *
Reason: We can use the negative exponent definition to write f(x) = ( 12)^x as f(x) = 2^x for negative exponent values. Note than the bigger the negative value x takes, the bigger the value the function takes.
The domain of the function is all real numbers. ✓
Reason: A power with a positive base is defined for any exponent value.
The range of the function is all real numbers greater than zero. ✓
Reason: A power with a positive base cannot take negative values, nor be 0.
When x=0, the value of the function equals the coefficient multiplying the power. In this case, f(0) = 1. ✓
Reason: For any nonzero number a, a^0=1. For this case, f(0) = ( 12)^0 = 1.
e
The graph of the function f(x) = 3( 12)^x is shown below.
Now, let's check which characteristics this function shows and analyze the function's form to understand why.
As x takes bigger negative values, the function approaches 0. *
Reason: We can use the negative exponent definition to write f(x) = 3( 12)^x as f(x) = 3(2)^x for negative exponent values. Note than the bigger the negative value x takes, the bigger the value the function takes.
The domain of the function is all real numbers. ✓
Reason: A power with a positive base is defined for any exponent value.
The range of the function is all real numbers greater than zero. ✓
Reason: A power with a positive base cannot take negative values, nor be 0.
When x=0, the value of the function equals the coefficient multiplying the power. In this case, f(0) = 3. ✓
Reason: For any nonzero number a, a^0=1. For this case, f(0) = 3( 12)^0 = 3(1)=3.
f
The graph of the function f(x) = 2( 34)^x, is shown below.
Now, let's check which characteristics does this function show and analyze the function's form to understand why.
As x takes bigger negative values, the function approaches 0. *
Reason: We can use the negative exponent definition to write f(x) = 2( 34)^x as f(x) = 2( 43)^x for negative exponent values. Note that the bigger the negative value x takes, the bigger the value the function takes.
The domain of the function is all real numbers. ✓
Reason: A power with a positive base is defined for any exponent value.
The range of the function is all real numbers greater than zero. ✓
Reason: A power with a positive base cannot take negative values, nor be 0.
When x=0, the value of the function equals the coefficient multiplying the power. In this case, f(0) = 3. ✓
Reason: For any nonzero number a, a^0=1. For this case, f(0) = 2( 34)^0 = 2(1)=2.
