Question

A4. Să se reprezinte grafic functiile definite prin:​

Answer 1

Explicație pas cu pas:

b) f(x) = |4x - 2| - x + 3 , x ∈ [-3; 4]

4x - 2 = 0, x = 1/2

|4x - 2| = 4x - 2, x ≥ 1/2

|4x - 2| = -4x + 2, x < 1/2

● f(x) = 4x - 2 - x + 3 = 3x + 1, x ∈ [1/2; 4]

f(0) = 1

f(x) = 0 => 3x + 1 = 0, x = -1/3

=> f(-1/3) = 0

f(1/2) = 5/2

● f(x) = -4x + 2 - x + 3 = -5x + 5, x ∈ [-3; 1/2)

f(0) = 5

f(x) = 0 => -5x + 5 = 0, x = 1

=> f(1) = 0

c) f(x) = |2 - x| - |2x|

x - 2 = 0 => x = 2

|x - 2| = x - 2, x ≥ 2

|x - 2| = -x + 2, x < 2

2x = 0 => x = 0

|2x| = 2x, x ≥ 0

|2x| = -2x, x < 0

● f(x) = x - 2 - 2x = -x - 2, x ∈ [2; ∞)

f(2) = -4 ; f(4) = -6

funcția este descrescătoare și nu intersectează axele de coordonate

● f(x) = -x + 2 - 2x = -3x + 2, x ∈ [0; 2)

f(0) = 2

f(x) = 0 => -3x + 2 = 0, x = 2/3

=> f(2/3) = 0

f(2) = -4

● f(x) = -x + 2 + 2x = x + 2, x ∈ (-∞; 0)

f(0) = 2

f(x) = 0 => x + 2 = 0, x = -2

=> f(-2) = 0

f(-4) = -2

d) f(x) = |x - 3| - | x + 1|, x ∈ [-4; 4]

x - 3 = 0 => x = 3

|x - 3| = x - 3, x ≥ 3

|x - 3| = -x + 3, x < 3

x + 1 = 0 => x = -1

| x + 1| = x + 1, x ≥ -1

| x + 1| = -x - 1, x < -1

● f(x) = x - 3 - x - 1 = -4, x ∈ [3; 4]

f(3) = -4; f(4) = -4

● f(x) = -x + 3 - x - 1 = -2x + 2, x ∈ [-1; 3)

f(0) = 2

f(x) = 0 => -2x + 2 = 0, x = 1

=> f(1) = 0

● f(x) = -x + 3 + x + 1 = 4, x ∈ [-4; -1)

f(-4) = 4; f(-1) = 4