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Wize High School Grade 12 Calculus Textbook > Review

Functions

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Functions

A function f(x)f\left(x\right) takes in a set of input xx values and produces a set of output values f(x)f\left(x\right).

Example
Given f(x)=3x2+4x+9f\left(x\right)=3x^2+\sqrt{4x+9}, evaluate the following:
a) f(0)=f\left(0\right)=
f(0)=3(0)2+4(0)+9f\left(0\right)=3\left(0\right)^2+\sqrt{4\left(0\right)+9}
f(0)=0+9f\left(0\right)=0+\sqrt{9}
f(0)=3f\left(0\right)=3

b) f(−1)=f\left(-1\right)=
f(−1)=3(−1)2+4(−1)+9f\left(-1\right)=3\left(-1\right)^2+\sqrt{4\left(-1\right)+9}
f(−1)=3+5f\left(-1\right)=3+\sqrt{5}

c) f(a)=f\left(a\right)=
f(a)=3a2+4a+9f\left(a\right)=3a^2+\sqrt{4a+9}

d) f(2+h)f\left(2+h\right)
f(2+h)=3(2+h)2+4(2+h)+9f\left(2+h\right)=3\left(2+h\right)^2+\sqrt{4\left(2+h\right)+9}
f(2+h)=3(4+4h+h2)+8+h+9f\left(2+h\right)=3\left(4+4h+h^2\right)+\sqrt{8+h+9}
f(2+h)=12+12h+3h2+h+9f\left(2+h\right)=12+12h+3h^2+\sqrt{h+9}

e) f(a+h)=f\left(a+h\right)=
f(a+h)=3(a+h)2+4(a+h)+9f\left(a+h\right)=3\left(a+h\right)^2+\sqrt{4\left(a+h\right)+9}
f(a+h)=3(a2+2ah+h2)+4a+4h+9f\left(a+h\right)=3\left(a^2+2ah+h^2\right)+\sqrt{4a+4h+9}
f(a+h)=3a2+6ah+3h2+4a+4h+9f\left(a+h\right)=3a^2+6ah+3h^2+\sqrt{4a+4h+9}

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Domain: The set of possible input xx values
Range: The set of possible output yy values

Example
Find the domain and range of the following functions:
a) y=2x+1y=2x+1
  • Domaine: we can input any xx value → D: x∈Rx\in R
  • Range: we can get any output value yy → R: y∈Ry\in R

b) y=(x−2)2+3y=(x-2)^2+3
  • Domain: we can input any xx value → D: x∈Rx\in R
  • Range: since (x−2)3≥0(x-2)^3\ge0, the yy value must be ≥3\ge3 → Range: y≥3y\ge3

c) x2+y2=4x^2+y^2=4
This is a circle with radius 2
  • D: −2≤x≤2-2\le x\le2
  • R: −2≤y≤2-2\le y\le2

Practice: Piecewise Functions

Sketch the following function and then evaluate the given function values.
f(x)={3x−2,x<1(x+1)2+2,x≥1f\left(x\right)= \begin{cases} 3x-2,&x<1\\ (x+1)^2+2,&x\ge1 \end{cases}

a) f(0)f(0)

b) f(1)f(1)

c) f(2)f(2)