given f(x)= 5x+4/5x^2+4x and g(x) = 1/x
a. determine the domains of f(x) and g(x)
b. simplify f(x) and find any vertical asymptotes
c. explain how the two functions differ
I have absolutely no idea how to answer this question - can you please help me to figure it out?!
If you really have no idea, you must not have looked at the material given you...
the domain of any rational function is where the denominator is zero
5x^2+4x = x(5x+4). so,
the domain of f is all reals except x = 0 or -4/5
the domain of g is all reals except x=0
I think you can simplify f(x).
Vertical asymptotes are where the denominator is zero and the numerator is not zero.
f(x) = 1/x except where x = -5/4
g(x) = 1/x
both are undefined at x=0
Of course! I'll explain each part step by step.
a. To determine the domains of functions, we need to find any values of x that would make the function undefined or result in division by zero. In the case of f(x), the denominator of the fraction is 5x^2 + 4x. For a fraction to be defined, the denominator cannot be zero. So, let's solve the equation 5x^2 + 4x = 0 to find any values that make the denominator zero.
Factorizing the equation, 5x(x + 4/5) = 0.
Setting each factor equal to zero, we have:
5x = 0 => x = 0
x + 4/5 = 0 => x = -4/5
Therefore, the values x = 0 and x = -4/5 make the denominator zero, so we can say that the domain of f(x) is all real numbers except x = 0 and x = -4/5.
For g(x), we have g(x) = 1/x. Here, the only value that would make the denominator zero is x = 0. So, the domain of g(x) is all real numbers except x = 0.
b. To simplify f(x), we need to perform the required operations. Given f(x) = (5x + 4)/(5x^2 + 4x), we can simplify as follows:
First, factor out the numerator: f(x) = 1(5x + 4)/(5x^2 + 4x).
Then, factor out the common term from the denominator: f(x) = (5x + 4)/(x(5x + 4)).
Now we can cancel out the common term: f(x) = 1/x.
As we have simplified f(x) to 1/x, its expression no longer contains any x terms in the denominator. Hence, there are no vertical asymptotes for f(x).
c. The two functions, f(x) = 5x + 4/(5x^2 + 4x) and g(x) = 1/x, differ in their expressions and the behavior of their graphs.
- Expression: f(x) is a rational function with a numerator of 5x + 4 and a denominator of 5x^2 + 4x. On the other hand, g(x) is a simple rational function with just the numerator 1 and the denominator x.
- Vertical Asymptotes: f(x) does not have any vertical asymptotes, whereas g(x) has a vertical asymptote at x = 0 due to its denominator being zero at that point.
- Domains: The domain of f(x) is all real numbers except x = 0 and x = -4/5, whereas the domain of g(x) is all real numbers except x = 0.
So, the two functions differ in terms of their expressions, presence of vertical asymptotes (only for g(x)), and their domains.