The features of function g if g(x)=f(x+4)+8 are
- y-intercept at (0,10)
- vertical asymptote of x=-4
How to determine features of function g?
The equation of function f is given as:
[tex]f(x)= \log_2(x)[/tex]
The function g is given as:
g(x) = f(x + 4) + 8
This means that:
[tex]g(x)= \log_2(x + 4) + 8[/tex]
Set the radicand to 0 to determine the domain and the vertical asymptote
x + 4 = 0
Solve for x
x = -4
This means that the domain is (-4, inf) and the vertical asymptote is x = -4
The function can output any real number.
So, the range is (-inf, inf)
Set x to 0 to determine the y-intercept
[tex]g(0)= \log_2(0 + 4) + 8[/tex]
Evaluate the sum
[tex]g(0)= \log_2(4) + 8[/tex]
Apply the law of logarithm
g(0)= 2 + 8
g(0) = 10
Hence, the y-intercept is (0,10)
Set g(x) to 0 to determine the x-intercept
[tex]\log_2(x + 4) + 8 = 0[/tex]
Subtract 8 from both sides
[tex]\log_2(x + 4) = -8[/tex]
Express as exponents
[tex]x + 4 = 2^{-8[/tex]
This gives
x + 4 = 0.00390625
Subtract 4 from both sides
x = -3.99609375
Approximate
x = -4
Hence, the x-intercept is (-4,0)
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