Hi! I really need help with these. I tried them on my own but I got really confused. Thanks!
1.) If f(x)=1/x and g(x)=1/x^2, find the domain of g/f
2.) If h(x)=4/(5x+2)^2, find the two functions f and g so that f(g(x))=h(x).
1. keep in mind that both f(x) and g(x) would be undefined for x=0
so even though g/f = (1/x) / (1/x^2) = x
the final function would still be undefined (actually it would be indeterminate) for x = 0
so the domain is :
set of all real numbers, x ≠ 0
183.79
Hello! I'd be happy to help you with these questions. Let's start with the first one.
1.) To find the domain of g/f, we need to determine the values of x where both the numerator and denominator are defined and not equal to zero.
The function f(x) is defined for all real numbers except x = 0 because dividing by zero is undefined.
The function g(x) is defined for all real numbers except x = 0 as well because dividing by zero is undefined.
Therefore, the domain of g/f would be all real numbers except x = 0 since both f(x) and g(x) are undefined at x = 0.
Now let's move on to the second question.
2.) To find the functions f and g such that f(g(x)) = h(x), we need to perform function composition.
Given h(x) = 4/(5x+2)^2, we can see that h(x) is a complex expression involving a quadratic term.
To simplify this function, we can follow these steps:
Step 1: Let g(x) = 5x + 2.
We choose this value for g(x) because it allows us to cancel out the numerator of h(x).
Step 2: Now we need to find f(g(x)). Let's substitute the expression for g(x) into f(x). This gives us f(5x + 2).
Step 3: Finally, we set f(g(x)) = h(x). So, f(5x + 2) = 4/(5x + 2)^2.
In summary, the functions f(x) = 4/x^2 and g(x) = 5x + 2 satisfy f(g(x)) = h(x) = 4/(5x + 2)^2.
I hope this helps! Let me know if you have any further questions.