Answer:
- f(x) has the greater y-intercept
- It is greater than that of g(x) by 4 units
Step-by-step explanation:
- The equation for an exponential function f(x) is given by the formula:
[tex]f(x) = a \cdot b^x[/tex] [1]
where
a, b are constants
- Let's examine the table giving f(x) for various values of x and determine first its equation and thereby its y-intercept
- We note that at x = 1, f(x) = 12 so f(1) = 12
Plugging these values into the exponential expression for f(x) we get
[tex]f(1) = a \cdot b^1 = ab[/tex]
[tex]ab = 12\\[/tex] [2]
- At x = 2, f(x) = 6 or equivalently f(2) = 6
[tex]f(2) = a \cdot b^2[/tex]
[tex]a \cdot b^2 = 6[/tex] [3]
- Divide eq [3] by eq [2]:
[tex]\dfrac{ab^2}{ab} = \dfrac{6}{12}= 0.5[/tex]
[tex]= > b = 0.5[/tex]
- Substitute for b in equation [2], ab = 12 to get
[tex]a (0.5) = 12\\\\a = \dfrac{12}{0.5}\\\\a = 24[/tex]
- Therefore,
[tex]f(x) = 24 (0.5)^x[/tex]
- The y-intercept occurs at x = 0
Substitute x = 0 in the expression for f(x):
[tex]f(0) = 24 (0.5)^0 = 24 \cdot 1 = 24[/tex]
- Therefore the y-intercept for f(x) = 24
- g(x) is also an exponential function that increases by a factor of 2 for every unit increase in x
- This can be modeled as:
[tex]g(x) = a \cdot 2^x[/tex]
- Since the graph of g(x) passes through the point (2, - 5) this means
at x = -2, f(x) = 5
or
[tex]g(-2) = 5[/tex]
- Using the expression for g(x) at x = - 2 gives
[tex]g(-2) = a \cdot 2^{-2}\\\\[/tex]
[tex]= > a \cdot 2^{-2} = 5\\\\[/tex]
[tex]= > a \cdot \dfrac{1}{2^2} = 5\\\\[/tex]
[tex]= > a \cdot \dfrac{1}{4} = 5\\\\[/tex]
[tex]= > a = 4 \cdot 5 \\\\a = 20[/tex]
- Equation for g(x) is
[tex]g(x) = 20 \cdot 2 ^ x[/tex]
- y-intercept is at x = 0
[tex]= > g(0) = 20 \cdot 2 ^0\\\\= > g(0) = 20 \cdot 1\\\\= > g(0) = 20\\\\[/tex]
- Therefore y-intercept of g(x) = 20
Answer
f(x) has the greater y-intercept
It is greater than that of g(x) by 4 units
