5. For the log function (h(x)=log(x+3)-8):
a) Find the domain.
b) Find the asymptotes.
c) Find the x-intercepts.
My job is not to just do it for you but to help if you get stuck.
note that this is just log(x) shifted left 3 and down 8.
So, now you can relate its properties to those of h(x)
To find the domain, asymptotes, and x-intercepts of the given log function, let's break it down step by step:
a) Finding the domain:
The domain of a logarithmic function is the set of all possible input values that the function can take. For a logarithm function, the input value must be greater than 0 since the logarithm of a negative number or zero is undefined.
In this case, we have h(x) = log(x + 3) - 8. Since log(x + 3) requires x + 3 to be greater than 0, we set up the inequality:
x + 3 > 0
Solving this inequality, we subtract 3 from both sides:
x > -3
Therefore, the domain of the function h(x) is all real numbers greater than -3.
b) Finding the asymptotes:
Asymptotes are lines that the graph of a function approaches but never touches. For logarithmic functions of the form log(x + a), the vertical asymptote is given by x = -a.
In this case, the formula is h(x) = log(x + 3) - 8. Therefore, the vertical asymptote is x = -3 (since a = -3).
c) Finding the x-intercepts:
To find the x-intercepts, we need to find the values of x for which h(x) equals zero (i.e., where the graph crosses the x-axis).
In this case, h(x) = log(x + 3) - 8. We set h(x) = 0 and solve for x:
log(x + 3) - 8 = 0
Adding 8 to both sides of the equation:
log(x + 3) = 8
Since the logarithm function is the inverse of exponential function, we rewrite the equation in exponential form:
10^8 = x + 3
Simplifying:
100,000,000 = x + 3
Subtracting 3 from both sides:
x = 100,000,000 - 3
Therefore, the x-intercept of the function h(x) is x = 99,999,997.