For the log function (h(x)=log(x+3)-8):
a) Find the domain.
b) Find the asymptotes.
c) Find the x-intercepts.
To find the domain of a logarithmic function, we need to consider the restrictions on the input value, which is the x-value in this case.
a) Domain:
In general, for a logarithmic function, the argument (inside the logarithm) must be greater than zero. Here, we have h(x) = log(x+3) - 8.
For the argument x+3 to be greater than zero, we need x+3 > 0. Solving this inequality, we get x > -3. Therefore, the domain of the function h(x) = log(x+3) - 8 is all real numbers greater than -3.
b) Asymptotes:
The asymptotes of a logarithmic function are vertical lines that the graph approaches but never touches. For a logarithmic function in the form h(x) = log(x+a) + b, the vertical asymptote is given by x = -a.
In this case, we have h(x) = log(x+3) - 8. So, the vertical asymptote is x = -3.
c) X-intercepts:
To find the x-intercepts, we set the function h(x) equal to zero and solve for x.
0 = log(x+3) - 8
To isolate the logarithm, we add 8 to both sides:
8 = log(x+3)
Now, we can rewrite the equation in exponential form:
10^8 = x+3
Simplifying further, we get:
x + 3 = 100,000,000
Subtracting 3 from both sides:
x = 100,000,000 - 3
Therefore, the x-intercept of the function h(x) = log(x+3) - 8 is at x = 99,999,997.