Sure! Let's go through each function step-by-step and find the images of the given values [tex]\([-3, -1, 0, 1, 2, 4]\)[/tex].
### (a) [tex]\( f(x) = x^2 - x - 5 \)[/tex]
To find the images of [tex]\( f(x) \)[/tex], we will substitute the values into the function [tex]\( f(x) \)[/tex].
1. For [tex]\( x = -3 \)[/tex]:
[tex]\[
f(-3) = (-3)^2 - (-3) - 5 = 9 + 3 - 5 = 7
\][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = (-1)^2 - (-1) - 5 = 1 + 1 - 5 = -3
\][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 0^2 - 0 - 5 = -5
\][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 1^2 - 1 - 5 = 1 - 1 - 5 = -5
\][/tex]
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 2^2 - 2 - 5 = 4 - 2 - 5 = -3
\][/tex]
6. For [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = 4^2 - 4 - 5 = 16 - 4 - 5 = 7
\][/tex]
So, the images of [tex]\( f(x) \)[/tex] for [tex]\([-3, -1, 0, 1, 2, 4]\)[/tex] are:
[tex]\[
\boxed{[7, -3, -5, -5, -3, 7]}
\][/tex]
### (b) [tex]\( g(x) = (x+1)^2 \)[/tex]
To find the images of [tex]\( g(x) \)[/tex], we will substitute the values into the function [tex]\( g(x) \)[/tex].
1. For [tex]\( x = -3 \)[/tex]:
[tex]\[
g(-3) = (-3+1)^2 = (-2)^2 = 4
\][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[
g(-1) = (-1+1)^2 = 0^2 = 0
\][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = (0+1)^2 = 1^2 = 1
\][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = (1+1)^2 = 2^2 = 4
\][/tex]
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = (2+1)^2 = 3^2 = 9
\][/tex]
6. For [tex]\( x = 4 \)[/tex]:
[tex]\[
g(4) = (4+1)^2 = 5^2 = 25
\][/tex]
So, the images of [tex]\( g(x) \)[/tex] for [tex]\([-3, -1, 0, 1, 2, 4]\)[/tex] are:
[tex]\[
\boxed{[4, 0, 1, 4, 9, 25]}
\][/tex]
### (c) [tex]\( h(x) = x - 1 \)[/tex]
To find the images of [tex]\( h(x) \)[/tex], we will substitute the values into the function [tex]\( h(x) \)[/tex].
1. For [tex]\( x = -3 \)[/tex]:
[tex]\[
h(-3) = -3 - 1 = -4
\][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[
h(-1) = -1 - 1 = -2
\][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[
h(0) = 0 - 1 = -1
\][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[
h(1) = 1 - 1 = 0
\][/tex]
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[
h(2) = 2 - 1 = 1
\][/tex]
6. For [tex]\( x = 4 \)[/tex]:
[tex]\[
h(4) = 4 - 1 = 3
\][/tex]
So, the images of [tex]\( h(x) \)[/tex] for [tex]\([-3, -1, 0, 1, 2, 4]\)[/tex] are:
[tex]\[
\boxed{[-4, -2, -1, 0, 1, 3]}
\][/tex]
### (d) [tex]\( F(x) = (x+1)(x-2) \)[/tex]
To find the images of [tex]\( F(x) \)[/tex], we will substitute the values into the function [tex]\( F(x) \)[/tex].
1. For [tex]\( x = -3 \)[/tex]:
[tex]\[
F(-3) = (-3+1)(-3-2) = (-2)(-5) = 10
\][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[
F(-1) = (-1+1)(-1-2) = 0 \cdot (-3) = 0
\][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[
F(0) = (0+1)(0-2) = 1 \cdot (-2) = -2
\][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[
F(1) = (1+1)(1-2) = 2 \cdot (-1) = -2
\][/tex]
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[
F(2) = (2+1)(2-2) = 3 \cdot 0 = 0
\][/tex]
6. For [tex]\( x = 4 \)[/tex]:
[tex]\[
F(4) = (4+1)(4-2) = 5 \cdot 2 = 10
\][/tex]
So, the images of [tex]\( F(x) \)[/tex] for [tex]\([-3, -1, 0, 1, 2, 4]\)[/tex] are:
[tex]\[
\boxed{[10, 0, -2, -2, 0, 10]}
\][/tex]
