Two functions f and g are defined over the set of R of real numbers as follows: f:x=2x-3 g:x=xsquare+5 find f[g(x-1)]
f(x) = 2x-3
so,
f(g) = 2g-3
since g = x^2+5,
g(x-1) = (x-1)^2+5
so,
f(g(x-1)) = 2((x-1)^2+5)-3 = 2x^2-4x+9
To find f[g(x-1)], we need to substitute g(x-1) into the function f.
First, let's find g(x-1).
Given:
g(x) = x^2 + 5
Substitute x-1 for x:
g(x-1) = (x-1)^2 + 5
Simplify:
g(x-1) = x^2 - 2x + 1 + 5
g(x-1) = x^2 - 2x + 6
Now that we have the expression for g(x-1), let's substitute it into the function f.
Given:
f(x) = 2x - 3
Substitute g(x-1) for x:
f[g(x-1)] = 2(g(x-1)) - 3
Substitute x^2 - 2x + 6 for g(x-1):
f[g(x-1)] = 2((x^2 - 2x + 6)) - 3
Simplify:
f[g(x-1)] = 2x^2 - 4x + 12 - 3
f[g(x-1)] = 2x^2 - 4x + 9
Therefore, f[g(x-1)] = 2x^2 - 4x + 9.
To find the value of f[g(x-1)], we need to substitute (x-1) into the function g(x) and then substitute the result into the function f(x).
First, let's find the value of g(x-1):
g(x) = x^2 + 5
Substitute (x-1) into g(x):
g(x-1) = (x-1)^2 + 5
= (x^2 - 2x + 1) + 5
= x^2 - 2x + 6
Now that we have g(x-1), let's substitute it into f(x):
f(x) = 2x - 3
Substitute g(x-1) into f(x):
f[g(x-1)] = 2(g(x-1)) - 3
= 2(x^2 - 2x + 6) - 3
= 2x^2 - 4x + 12 - 3
= 2x^2 - 4x + 9
Therefore, f[g(x-1)] = 2x^2 - 4x + 9.