Find two functions, f and g, to express the given function (y = [4/(x-3)] + 1) in each way shown below.
Sum
Product
Quotient
Composition
I got the sum and the composition, but I can't figure out the product and the quotient.
Any help is appreciated, thanks!
To find the product of two functions, let's denote the given function as y = f(x) = 4/(x-3) + 1. Now we need to find another function g(x) such that the product of f(x) and g(x), denoted as h(x) = f(x) * g(x), is equal to y.
To find g(x) for the product, we can start by expressing the given function as y = f(x) = 4/(x-3) + 1. Let's manipulate the equation to solve for g(x):
y = f(x) = 4/(x-3) + 1
Subtract 1 from both sides:
y - 1 = 4/(x-3)
Now, let's express the right side of the equation using the desired product form f(x) * g(x):
4/(x-3) = f(x) * g(x)
To find g(x), we need to isolate it on one side of the equation. Multiply both sides by (x-3):
4 = (x-3) * (f(x) * g(x))
Now divide both sides by f(x) * (x-3):
4 / (f(x) * (x-3)) = g(x)
So, the function g(x) for the product is:
g(x) = 4 / (f(x) * (x-3))
Now, let's move on to finding the quotient of two functions.
To find the quotient of two functions, we need to express the given function f(x) = 4/(x-3) + 1 as a division of two functions, f(x) = u(x) / v(x). We will denote the quotient as h(x) = f(x) / g(x) = (u(x) / v(x)) / g(x), where g(x) is the second function.
To find g(x) for the quotient, we can rewrite the given function as a division:
f(x) = 4/(x-3) + 1 = (4 + (x-3))/(x-3) = (x + 1)/(x-3)
Now, in order to find g(x), we need to express the right side of the equation as a division of two functions:
(x + 1)/(x-3) = (u(x) / v(x))
In this case, the numerator u(x) is x + 1 and the denominator v(x) is x - 3. Therefore, the function g(x) for the quotient is:
g(x) = (u(x) / v(x)) = (x + 1)/(x-3)
I hope this explanation helps you understand how to find the product and quotient of functions!
To find the product and quotient of the function y = [4/(x-3)] + 1, we need to express it in terms of two separate functions.
Product:
Let's express y as the product of two functions, f and g.
y = f(x) * g(x)
To find f(x) and g(x), let's rewrite the original function:
y = [4/(x-3)] + 1
Now, let's express it as a product:
y = [4/(x-3)] * [1 + (x-3)/(x-3)]
= [4/(x-3)] * [(x-3)/(x-3) + 1]
= [4/(x-3)] * [(x-3 + x-3)/(x-3)]
= [4/(x-3)] * [(2x-6)/(x-3)]
= 4(2x-6)/(x-3)^2
Therefore, f(x) = 4 and g(x) = (2x-6)/(x-3)^2.
Quotient:
Let's express y as the quotient of two functions, f and g.
y = f(x) / g(x)
To find f(x) and g(x), let's rewrite the original function:
y = [4/(x-3)] + 1
Now, let's express it as a quotient:
y = [4/(x-3)] / [1/(x-3)]
= [4/(x-3)] * [(x-3)/1]
= 4
Therefore, f(x) = 4 and g(x) = 1.
In summary:
Product: y = f(x) * g(x)
f(x) = 4
g(x) = (2x-6)/(x-3)^2
Quotient: y = f(x) / g(x)
f(x) = 4
g(x) = 1