Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.
f(x) = x2 - 3 and g(x) = square root of quantity three plus x
I see a whole bunch of posts, but no indication of your attempts to solve them. I'll do one of them here:
f(g) = g^2-3 = (√(3+x))^2 - 3 = 3+x-3 = x
g(f) = √(f+3) = √(x^2-3+3) = √x^2 = x
However, note that √x^2 is not necessarily x.
If x = -3, √9 = 3, not -3!
To confirm that f and g are inverses, we need to show that when we apply one function after the other, we get back the original input. Let's start with f(g(x)):
f(g(x)) = f(sqrt(3 + x))
To find the value of f(g(x)), we substitute g(x) into the function f. Since g(x) is the square root of (3 + x), we can write:
f(g(x)) = f(sqrt(3 + x)) = (sqrt(3 + x))^2 - 3 = 3 + x - 3 = x
As you can see, f(g(x)) simplifies to x. This indicates that applying function g followed by f gives us back the original input x.
Now let's move on to g(f(x)):
g(f(x)) = g(x^2 - 3)
To find the value of g(f(x)), we substitute f(x) into the function g. Since f(x) is x^2 - 3, we can write:
g(f(x)) = g(x^2 - 3) = sqrt(3 + (x^2 - 3)) = sqrt(x^2) = |x|
In this case, g(f(x)) simplifies to |x|, which is the absolute value of x. As you can see, applying function f followed by g does not give us back the original input x, so these functions are not inverses of each other.
To confirm that f and g are inverses, we need both f(g(x)) = x and g(f(x)) = x to hold. In this case, only f(g(x)) = x holds, so f and g are not inverses of each other.