Thanks steve for a brilliant answer! Got one last one which is gobbledegook to me.
Two functions f and g are defined by
f:x(arrow)x^2 + 3
g:x(arrow)2x + 1
find:
a) the function fg
b)solve the equation f(x) = 12g^-1(x)
Many Thanks!!!!
using the usual notation,
f(x) = x^2+3
g(x) = 2x+1
fg = f(x)*g(x) = (x^2+3)(2x+1) = 2x^3 + x^2 + 6x + 3
f(x) = 12g^-1(x)
g^-1(x) = (x-1)/2
so, solve x^2+3 = 12(x-1)/2
x^2+3 = 6x-6
x^2 - 6x + 9 = 0
(x-3)^2 = 0
x = 3
If by fg you meant f◦g (that is, f(g(x))), then
since f(x) = x^2+3,
f(g) = g^2+3 = (2x+1)^2+3 = 4x^2 + 4x + 4
To find the function fg, you have to substitute the function g into the function f. This means that wherever there is an 'x' in function f, you replace it with the expression for g(x).
a) Let's find fg(x):
f(x) = x^2 + 3
g(x) = 2x + 1
Replacing x in f(x) with g(x), we have:
fg(x) = f(g(x))
= (2x + 1)^2 + 3
Expanding, we get:
fg(x) = 4x^2 + 4x + 1 + 3
= 4x^2 + 4x + 4
So, the function fg is given by fg(x) = 4x^2 + 4x + 4.
b) To solve the equation f(x) = 12g^-1(x), we need to find the inverse of g(x) and substitute it into f(x).
First, let's find g^-1(x), the inverse of g(x):
g(x) = 2x + 1
To find the inverse, swap x and y and solve for y:
x = 2y + 1
2y = x - 1
y = (x - 1)/2
So, the inverse of g(x), g^-1(x), is given by g^-1(x) = (x - 1)/2.
Now, let's solve f(x) = 12g^-1(x):
f(x) = x^2 + 3
Replacing g^-1(x) with (x - 1)/2, we have:
x^2 + 3 = 12((x - 1)/2)
Simplifying:
x^2 + 3 = 6x - 6
x^2 - 6x + 9 = 0
(x - 3)^2 = 0
Taking the square root:
x - 3 = 0
x = 3
So, the solution to the equation f(x) = 12g^-1(x) is x = 3.