1.07 - Piecewise Functions | High School

University:
High School
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High School Physics
Academic year:

2024
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10
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1.07 Piecewise Functions • Piecewise Functions are composed of restricted pieces of other functions The Concept Any other function that you have studied: • Linear • Quadratic • GIF • Absolute • Square Root • Rational • Exponential (next unit) • Logarithmic (term 2) • Sinusoidal (term 2) Pieces of What? 2𝑥 + 3 𝑜𝑣𝑒𝑟 ] − ∞, 10] 2 4 𝑥 − 11 + 19 10, 16 𝑓 𝑥 =൞ −7 𝑥 − 16 + 119 [16, ∞[ - Function 𝑓(𝑥) is composed of three pieces of functions which change types at 10 and 16 on the x-axis - Transition points may inclusive to both functions on either side, or inclusive to only one side and exclusive on the other (open circles). The Notation - Example 2𝑥 + 3 𝑜𝑣𝑒𝑟 ] − ∞, 10] 10, 16 𝑓 𝑥 = ൞4 𝑥 − 11 2 + 19 −7 𝑥 − 16 + 119 [16, ∞[ What this looks like 2𝑥 + 3 𝑜𝑣𝑒𝑟 ] − ∞, 10] 10, 16 𝑓 𝑥 = ൞4 𝑥 − 11 2 + 19 −7 𝑥 − 16 + 119 [16, ∞[ Find 𝑓 −9 𝑓 −9 = 2 −9 + 3 = −15 Solving Example 1 ← −9 2𝑥 + 3 𝑜𝑣𝑒𝑟 ] − ∞, 10] 10, 16 𝑓 𝑥 = ൞4 𝑥 − 11 2 + 19 −7 𝑥 − 16 + 119 [16, ∞[ Find 𝑓 16 𝑓 16 = 4 16 − 11 2 + 19 = 119 𝑓 16 = −7 16 − 16 + 119 = 119 Solving Example 2 ← 16 ← 16 2𝑥 + 3 𝑜𝑣𝑒𝑟 ] − ∞, 10] 10, 16 𝑓 𝑥 = ൞4 𝑥 − 11 2 + 19 −7 𝑥 − 16 + 119 [16, ∞[ 𝑓 𝑥 = 22 𝑓 𝑥 = 22 Find 𝑓 𝑥 = 22 4 𝑥 − 11 2 + 19 = 22 𝑓 𝑥 = 22 𝑥 − 11 2𝑥 + 3 = 22 2𝑥 = 19 𝑥= 19 2 2 3 = 4 𝑥 − 16 = 22− 3 2 𝑥3 = 97 7 9409 𝑥 − 16 = 49 3 𝑥 − 11 = ± 2 𝑥2 = −7 𝑥 − 16 + 119 = 22 22+ 3 2 Solving Example 3 𝑥4 = 10193 49 2𝑥 + 3 𝑜𝑣𝑒𝑟 ] − ∞, 10] 10, 16 𝑓 𝑥 = ൞4 𝑥 − 11 2 + 19 −7 𝑥 − 16 + 119 [16, ∞[ 𝑓 𝑥 = 40 𝑓 𝑥 = 40 Find 𝑓 𝑥 = 40 4 𝑥 − 11 2 + 19 = 40 𝑓 𝑥 = 40 𝑥 − 11 2𝑥 + 3 = 40 2𝑥 = 37 𝑥= 37 2 2 21 = 4 𝑥 − 16 = 22− 21 2 𝑥2 = 79 7 6241 𝑥 − 16 = 49 21 𝑥 − 11 = ± 2 𝑥1 = −7 𝑥 − 16 + 119 = 40 22+ 21 2 Solving Example 4 𝑥1 = 7025 49 (𝑥 + 2)2 −5 ≤ 𝑥 ≤ 1 • 𝑓 𝑥 = ൞−2 𝑎𝑥 − 2 + 9 1 ≤ 𝑥 ≤ 3 − 𝑥 − 6 + 𝑏 3 ≤ 𝑥 ≤ 10 𝒇 𝟏𝟎 = ? f 1 = (1 + 2)2 = 9 𝑓 1 = −2 𝑎(1) − 2 + 9 = 9 −2 𝑎 − 2 = 0 𝑎=2 𝑓 3 = −2 2 3 − 2 + 9 = 5 𝑓 3 =− 3−6 +𝑏 =5 −3 + 𝑏 = 5 𝑏=8 Example 5 𝑓 10 = − 10 − 6 + 8 𝑓 10 = 4