Answer:
Both f(x) and g(x) have a common domain on the interval (0, ∞).
Step-by-step explanation:
Given functions:
[tex]\begin{cases}f(x)=\log_2(4x)\\ g(x)=4^x-3 \end{cases}[/tex]
The domain of a function is the set of all possible input values (x-values).
Since we cannot take logs of negative numbers or zero, the domain of function f(x) is (0, ∞).
The domain of function g(x) is unrestricted and therefore (-∞, ∞).
Therefore, both functions have a common domain on the interval (0, ∞) since the domain (0, ∞) is part of (-∞, ∞).
The range of a function is the set of all possible output values (y-values).
The range of f(x) is unrestricted and therefore (-∞, ∞).
The parent function of g(x) is [tex]y=4^x[/tex]. This has a range of (0, ∞) as it has an asymptote at y = 0. Therefore, as g(x) is translated 3 units down, g(x) will have an asymptote at y = 3.
Therefore, the range of g(x) is (-3, ∞).
So f(x) and g(x) do not have the same range.
The x-intercept is when the curve crosses the x-axis, so when y = 0.
To find if the x-intercept of both functions is at x = 2, substitute this value into the functions and solve for y:
[tex]\begin{aligned}f(2) & =\log_2(4 \cdot 2)\\ & = \log_28\\ & = \log22^3\\ & = 3\log_22\\ & =3\end{aligned}[/tex]
[tex]\begin{aligned}g(2) & =4^2-3\\ & = 4 \cdot 4 - 3\\& =16-3\\ & =13\end{aligned}[/tex]
Therefore, f(x) and g(x) do not have an x-intercept at x = 2.
As the domain of f(x) is (0, ∞), it is undefined over the interval (-4, 0] and therefore is not increasing on the interval (-4, ∞).
Answer:
Option A
Step-by-step explanation:
#Domain
f(x) is logarithm function so domain is positive
g(x) has the domain R i e(-oo,oo)
So they both have common domain as (0,oo)
#Range
f(x) has domain R
g(x) is 4^x-3
Asymptote is present at y=0-3 i e y=3
So range is
Not same range
#x intercept
log_2(4x):-
4^x-3
Not same
#y intercept
First one is logarithm function, -4 is not in its domain so undefined .
