Mth101 function formula

Functions – Formula Sheet:
Average Rate of Change: The Difference Quotient:

𝑓(𝑏) − 𝑓(𝑎) 𝑓(𝑥 + ℎ) − 𝑓(𝑥)
𝑏−𝑎 ℎ

Vertical Line Test: Horizontal Line Test:

If a vertical line intersects a graph at more than If a horizontal line intersects a graph at only one
one point, then the relation does not represent a point, then the function is one-to-one. In
function. addition, the inverse function is also a function.

Even Functions: Odd Functions:

𝑓(−𝑥) = 𝑓(𝑥) 𝑓(−𝑥) = −𝑓(𝑥)

Composite Functions: Inverse Functions:

(𝑓 𝑜 𝑔)(𝑥) = 𝑓(𝑔(𝑥)) 𝐼𝑓 𝑓(𝑔(𝑥)) = 𝑥 𝑎𝑛𝑑 𝑔(𝑓(𝑥)) = 𝑥, 𝑡ℎ𝑒𝑛 ….

(𝑔 𝑜 𝑓)(𝑥) = 𝑔(𝑓(𝑥)) 𝑔(𝑥) = 𝑓 −1 (𝑥) 𝑎𝑛𝑑 𝑓(𝑥) = 𝑔−1 (𝑥)

The Distance Formula: The Midpoint Formula:

𝐷 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 𝑥1 + 𝑥2 𝑦1 + 𝑦2
𝑀( , )
2 2
Transformations – Vertical Shifts: Transformations – Horizontal Shifts:

𝑦 = 𝑓(𝑥) + 𝑐 𝑠ℎ𝑖𝑓𝑡 𝑢𝑝 𝑦 = 𝑓(𝑥 + 𝑐) 𝑆ℎ𝑖𝑓𝑡 𝑙𝑒𝑓𝑡
𝑦 = 𝑓(𝑥) − 𝑐 𝑠ℎ𝑖𝑓𝑡 𝑑𝑜𝑤𝑛 𝑦 = 𝑓(𝑥 − 𝑐) 𝑆ℎ𝑖𝑓𝑡 𝑟𝑖𝑔ℎ𝑡

Vertical Stretch: Horizontal Shrink:

𝑦 = 𝑐 ∙ 𝑓(𝑥) 𝑐>1 𝑦 = 𝑓(𝑐𝑥) 𝑐>1

Vertical Shrink: Horizontal Stretch:

𝑦 = 𝑐 ∙ 𝑓(𝑥) 0<𝑐<1 𝑦 = 𝑓(𝑐𝑥) 0<𝑐<1

Reflection about the x-axis: Reflection about the y-axis:

𝑦 = −𝑓(𝑥) 𝑦 = 𝑓(−𝑥)

Reflection about the origin: Reflection about the line y=x:

𝑦 = −𝑓(−𝑥) (𝑥, 𝑦) → (𝑦, 𝑥)
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