Let us get working :)
Given are f(x) = 2x - 1 and g(x) = x² + 2
i) fg(-2)
g(-2) = (-2)² + 2
= 4 + 2
= 6
↓ we use the value found to continue
f(6) = 2(6) - 1
= 12 - 1
= 11
Your final answer will be 11
ii) gf([tex] \frac{1}{2} [/tex])
f([tex] \frac{1}{2} [/tex]) = 2[tex] \frac{1}{2} [/tex]) - 1
= 1 - 1
= 0
↓
g(0) = (0)² + 2
= 2
Your final answer will be 2
iii)fg(x)
g(x) = x² + 2
↓
f(x² + 2) = 2(x² + 2) - 1
= 2x² + 4 - 1
= 2x² + 3
Your final asnwer will be 2x² + 3
iv) gf(x)
f(x) = 2x - 1
↓
g(x) = (2x - 1)² + 2
= (2x - 1)(2x - 1) + 2 (open brackets)
= 4x² - 4x + 1 + 2
= 4x² - 4x + 3
Your final answer will be 4x² - 4x + 3
For what values of x is gf(x) = fg(x)
We did these functions above so we substitute
4x² - 4x + 3 = 2x² + 3
-2x² - 3 -2x² - 3 (to take all to one side)
2x² - 4x = 0 (we equate the function to 0)
2x(x - 2) = 0
2x = 0 x - 2 = 0
x = 0 x = 2
When x = 0 or x = 2, fg(x) = gf(x)
