f(x)= x^2+5 g(x)= square root x-1
(f/g)(x)
Substitute sqrt(x-1)for x in the f(x) equation. You get
f{g(x)} = x-1 + 5 = x + 4
To find (f/g)(x), we need to divide f(x) by g(x).
First, we will replace f(x) with its expression: f(x) = x^2 + 5.
Next, we will replace g(x) with its expression: g(x) = √(x - 1).
Now, we can write the expression for (f/g)(x): (f/g)(x) = (x^2 + 5) / √(x - 1).
Therefore, the expression for (f/g)(x) is (x^2 + 5) / √(x - 1).
To find the quotient of two functions (f/g)(x), we need to divide the values of f(x) by g(x) for the same input value, x. In this case, f(x) is given by f(x) = x^2 + 5 and g(x) is given by g(x) = √(x-1).
To find the quotient (f/g)(x), substitute f(x) and g(x) into the expression and divide:
(f/g)(x) = f(x) / g(x) = (x^2 + 5) / (√(x-1))
Now, when simplifying this expression, it's important to note that the denominator contains a square root. To eliminate it, we need to multiply both the numerator and denominator by the conjugate of the denominator, which is (√(x-1) + 1).
(f/g)(x) = (x^2 + 5) / (√(x-1)) * (√(x-1) + 1) / (√(x-1) + 1)
Now, let's simplify further by expanding the numerator:
(f/g)(x) = (x^2 + 5)(√(x-1) + 1) / (√(x-1) * (√(x-1) + 1))
Using the distributive property, we can expand the numerator:
(f/g)(x) = (x^2 * √(x-1) + 5 * √(x-1) + x^2 + 5) / (√(x-1) * (√(x-1) + 1))
Now, let's simplify it one step further:
(f/g)(x) = (x^2 * √(x-1) + 5 * √(x-1) + x^2 + 5) / (x - 1 + √(x-1))
So, the quotient (f/g)(x) is given by (x^2 * √(x-1) + 5 * √(x-1) + x^2 + 5) / (x - 1 + √(x-1)).