for the given functions f and g find f o g and state the domain show all work f(x)=sqrt x-3 g(x)=x^2-7x+9
Usually f o g is defined as f(g(x))
f o g = f(x^2 - 7x + 9)
= √(x^2 - 7x + 9) - 3
if you meant to type f(x) = √(x-3) make the necessary adjustments, I read it just the way you typed it, i.e.
f(x) = √x - 3
Reiny,
that is correct, but I cannot type the squareroot sign with my computer - how do I find the domain, of this function?
the only thing to worry about for the domain is that the expression
x^2 - 7x + 9 > 0
look at the corresponding parabola
y = x^2 - 7x + 9
it has x-intercepts of
x = (7 ± √13)/2
so it will be negative between those two values
domain:
x < (7 - √13)/2 OR x > (7+√13)/2
To find f o g, which represents the composite function of f and g, you need to substitute g(x) into f(x). Here are the steps:
1. Start with the expression for g(x):
g(x) = x^2 - 7x + 9.
2. Substitute g(x) into f(x):
f(g(x)) = sqrt(g(x) - 3).
3. Substitute g(x) into the expression for f(g(x)):
f(g(x)) = sqrt((x^2 - 7x + 9) - 3).
4. Simplify the expression inside the square root:
f(g(x)) = sqrt(x^2 - 7x + 6).
Now, let's determine the domain of f(g(x)). The square root function is defined only for non-negative values inside the square root. Therefore, we need to find the values of x that make the expression inside the square root non-negative.
To do this, set the expression x^2 - 7x + 6 greater than or equal to zero and solve for x:
x^2 - 7x + 6 >= 0.
Factoring the quadratic equation gives:
(x-6)(x-1) >= 0.
Now, we can graph the inequality on a number line to find the range of x values that satisfy the inequality. The solution is x <= 1 or x >= 6.
Therefore, the domain of the composite function f(g(x)) is all real numbers except x values between 1 and 6 (excluding 1 and 6).
Domain: (-∞, 1] U [6, +∞)