Given the piece wise function f(x).
Question 1 :
Solve : f(x) = 1.
Answer :
Piece 1 :
x + 2 = 9
x = 7 ∈ x ≥ 2
Piece 2 :
x2 - 1 = 1
x2 = 2
x = ±√2
x ≈ -1.414 or 1.414
x = -1.414 is not in the interval 0 ≤ x < 2.
x ≈ 1.414 ∈ 0 ≤ x < 2
Piece 3 :
-e2x + 2 + 5 = 1
-e2x + 2 = -4
Multiply both sides by -1.
e2x + 2 = 4
2x + 2 = ln(4)
2x = ln(4) - 2
x = ½[ln(4) - 2]
x ≈ -0.307 ∈ x < 0
Question 2 :
Solve : f(x) = 4.
Answer :
Piece 1 :
x + 2 = 36
x = 34 ∈ x ≥ 2
Piece 2 :
x2 - 1 = 4
x2 = 5
x = ±√5
x ≈ ±2.236 ∉ 0 ≤ x < 2
NO SOLUTION
Piece 3 :
-e2x + 2 + 5 = 4
-e2x + 2 = -1
Multiply both sides by -1.
e2x + 2 = 1
2x + 2 = ln(1)
2x + 2 = 0
2x = -2
x = -1 ∈ x < 0
Question 3 :
Solve : f(x) = 7.
Answer :
Piece 1 :
x + 2 = 81
x = 79 ∈ x ≥ 2
Piece 2 :
x2 - 1 = 7
x2 = 8
x = ±√8
x ≈ ±2.828 ∉ 0 ≤ x < 2
NO SOLUTION
Piece 3 :
-e2x + 2 + 5 = 7
-e2x + 2 = 2
Soince a negative value ≠ a positive value, the above equation is false.
NO SOLUTION
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