Functions and Graphs

Function notation, composite and inverse functions, the shapes of the standard graphs, reading solutions off a curve, and differentiating to find stationary points. Functions and differentiation are Extended-only topics; the graph work is on both tiers.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ตรงกับ ม.4 ความสัมพันธ์และฟังก์ชัน และ ม.3 บทที่ 5 กราฟของฟังก์ชันกำลังสอง
Common Core (US): F-IF, F-BF
Cambridge: IGCSE 0580 — Topic 2: Algebra and graphs (Core and Extended)

5.1 Function notation (Extended)

นิยาม

Function

A rule giving exactly one output for each input. $f (x) = 3 x - 2$ means "multiply by three, then subtract two"; $f (4)$ is the output when the input is $4$ .

ตัวอย่าง

For $f (x) = 3 x - 2$ , find $f (4)$ and solve $f (x) = 13$ .

วิธีทำ

1. $f (4) = 12 - 2 = 10$
2. $3 x - 2 = 13$ , so $3 x = 15$

ตอบ $f (4) = 10$ and $x = 5$

Composite and inverse

fg (x) = f (g (x)) f (f^{- 1} (x)) = x

In $fg (x)$ the INNER function $g$ acts first. Read composites from the inside out.

ตัวอย่าง

With $f (x) = 2 x + 1$ and $g (x) = x^{2}$ , find $fg (3)$ and $f^{- 1} (x)$ .

วิธีทำ

1. $fg (3) = f (g (3)) = f (9) = 19$
2.For the inverse: write $y = 2 x + 1$ , swap to $x = 2 y + 1$
3.Solve for $y$ : $y = \frac{x - 1}{2}$

ตอบ $fg (3) = 19$ and $f^{- 1} (x) = \frac{x - 1}{2}$

ข้อควรระวัง

$f^{- 1} (x)$ is not $\frac{1}{f ( x )}$ , and $fg$ is not $gf$ . Both mistakes come from reading the notation as arithmetic when it is describing an operation.

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