Let (f) = 1/x and g(x)=x^2 +5x.
a. Find (f*g)(x)
b. Find the domain and range of (f*)(x)
I will interpret (f*g)(x) as f(g(x))
= f(x^2 + 5x)
= 1/(2x^2 + 5x)
clearly we cannot divide by zero.
When is 2x^2 + 5x equal to zero
x(2x + 5) = 0
x = 0, x = -5/2
so the domain would be any real number except x = 0 and x = -5/2
the range would be any real number, see the graph
https://www.wolframalpha.com/input/?i=plot+y+%3D+1%2F%282x%5E2+%2B+5x%29
Thank you so much :) This actually makes sense too.
To find the product of two functions (f*g)(x), you need to multiply the two functions together.
a. (f*g)(x) = f(x) * g(x) = (1/x) * (x^2 + 5x)
To simplify this, you can distribute the function f(x) = 1/x to the terms in g(x) = x^2 + 5x:
(f*g)(x) = (1/x) * x^2 + (1/x) * 5x
Now, simplify each term:
(f*g)(x) = x * x^2/x + 5x/x
(f*g)(x) = x^3/x + 5x/x
(f*g)(x) = x^2 + 5
Therefore, (f*g)(x) = x^2 + 5.
b. To find the domain and range of (f*)(x), first, let's clarify what is meant by (f*)(x). It seems like it might be a typo, as "*" typically represents multiplication, and the question previously asked for the product (f*g)(x). If the question is referring to the composition of the two functions f(x) and g(x), it would be written as (f ∘ g)(x) or (f ◦ g)(x).
Assuming that the question is asking for the domain and range of the composition (f ◦ g)(x), we proceed as follows:
The domain of a composition is determined by the domain of the inner function, g(x), in this case. Since g(x) = x^2 + 5x is a polynomial, it is defined for all real numbers. Therefore, the domain of (f ◦ g)(x) is also all real numbers.
To find the range, we need to examine the range of the outer function, f(x), which is f(x) = 1/x. The range of f(x) excludes x = 0 because division by zero is undefined. Hence, the range of (f ◦ g)(x) will also exclude the value zero.
Therefore, the domain of (f ◦ g)(x) is all real numbers except 0, and the range is all real numbers except 0.