Answer :

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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)





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