A function is defined as follows

(i) 2f(-4) + 3f(2)

(ii) f (- 7) - f (- 3)

(iii)

We are given,

f: [-7,6) → ℝ

[-7,6) = {x ∈ A: -7≤x<6}

= {-7, -6, -5, -4,-3, -2, -1, 0, 1, 2, 3, 4, 5}

And the function given are

(i) 2f(-4) + 3f(2)

Here the function lies between -5<x<2

So

f(x) = x + 5

f(-4) = -4 + 5

= 1

f(2) = 2 + 5

= 7

∴ 2f(-4) = 2 × 1 = 2

∴ 3f(2) = 3× 7 = 21

So 2f(-4) + 3f(2)

= 2 + 21 = 23

(ii) f (- 7) - f (- 3)

here the function lies between -7≤x<-5 and -5≤ x < 2

so f(x) = x² + 2x + 1

⇒ f(-7) = (-7)² + 2×(-7) + 1

= 49-14 + 1

= 36

Similarly f(x) = x + 5

⇒ f(-3) = -3 + 5

= 2

∴ f (- 7) - f (- 3)

= 36 -2

= 34

(iii) .

For the above statement the function lies between

-7≤ x <-5 for f(-6)

-5≤ x<2 for for f(-3) and f(1)

2<x<6 for f(4)

at f(x) = x² + 2x + 1 ; -7≤ x <-5

⇒ f(-6) = (-6)² + 2(-6) + 1

= 36-12 + 1 = 25

At f(x) = x + 5; -5≤ x<2 for

⇒ f(-3) = -3 + 5 = 2

⇒ 4f(-3) = 4 × 2 = 8

Similarly,

f(1) = 1 + 5 = 6

⇒ 3f(1) = 3 × 6 = 18

At f(x) = x-1; 2<x<6

⇒ f(4) = 4-1 = 3

⇒ 2f(4) = 2× 3 = 6

So the function given