A function is defined as follows

Find (i) f(5) + f(6) (ii) f(1)- f(-3)

(iii) f(-2h)- f(4) (iv) 

We are given,

f: [-3,7) → ℝ

And the function given is

(i) f(5) + f(6)

Here it is clear that 5 and 6 lies between 4 and 7 so we have to use the function,

f(x) = 2x - 3; 4 < x < 7

So,

f(5) = 2×5 - 3 = 10 - 3 = 7

f(6) = 2×6 - 1 = 12 - 3 = 9

∴ f(5) + f(6) = 7 + 9 = 16

(ii) f(1) - f(-3)

here its clear that 1 and -3 lies between the range of -3 ≤ x < 2 so we use the function

f(x) = 4x² - 1

⇒ f(1) = 4(1)²-1 = 4-1 = 3

⇒ f(-3) = 4(-3)²-1 = 4×9 - 1 = 35

So f(1) - f(-3)

= 3 - 35

= -32 

(iii) f(-2) - f(4)

f(x) = 4x² -1; -3 ≤ x <2

⇒ f(-2) = 4×(-2)²-1
           = 4×4 - 1
           = 16 - 1
           = 15

f(x) = 3x-2; 2 ≤ x ≤ 4

⇒ f(4) = 3 × 4 - 2
          = 12 - 2
          = 10

⇒ f(-2)- f(4) = 15-10 
                   = 5

(iv)

Here the entire values ranges from -1 to 6 so the range will lie between -3≤x<2, 2≤ x<4 and 4<x<7

So

f(3) = 3x - 2;  2≤ x<4
      = 3 × 3 – 2
      = 9 - 2 = 7

f(-1) = 4x² - 1: -3≤x<2
       = 4(-1)²-1
       = 4-1 = 3

f(6) = 2x - 3; 4<x<7
      = 2 × 6-3
      = 12-3 = 9

⇒ 2f(6) = 2 × 9 = 18

f(1) = 4x² - 1: -3≤x<2
      = 4(1)² - 1
      = 4 - 1 = 3
So the function