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Wize High School Grade 12 Pre-Calculus Textbook > Combination of Functions

Combining Families of Functions

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Combining Families of Functions

We can combine functions through addition, subtraction, multiplication, division, and composition as a way of creating a new functions with new behaviors.

Here is a list of common functions we have encountered:

Polynomial: y=axn+bx−1+...+cx+dy=ax^n+b^{x-1}+...+cx+d
Radical: y=nxy={}^n\sqrt{x}
Absolute Value: y=∣x∣y=|x|
Rational: y=P(x)Q(x),  where P(x) and Q(x) are polynomialsy=\dfrac{P(x)}{Q(x)},~~\text{where}~P(x)~\text{and}~Q(x)~\text{are polynomials}
Periodic: y=cos⁡θ, y=sin⁡θ, y=tan⁡θ, y=csc⁡θ, y=sec⁡θ, y=cot⁡θy=\cos\theta,~y=\sin\theta,~y=\tan\theta,~y=\csc\theta,~y=\sec\theta,~y=\cot\theta
Exponential: y=bxy=b^x
Logarithmic: y=log⁡bxy=\log_{b}x

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Even & Odd Functions

An even function has line symmetry and can be defined as:
f(−x)=f(x)\boxed{f(-x)=f(x)}

An odd function has point symmetry and can be defined as:
f(−x)=−f(x)\boxed{f(-x)=-f(x)}

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Maximum & Minimum Values

Given a function y=f(x)y=f(x) and some point (a, f(a))(a,~f(a)) is on the graph of y=f(x)y=f(x). Then,

A maximum value exists at x=ax=a if f(x)≤f(a)f(x)\leq{}f(a) for all values of x∈Rx\in\mathbb{R}.

A minimum value exists at x=ax=a if f(x)≥f(a)f(x)\geq{}f(a) for all values of x∈Rx\in\mathbb{R}.

Intervals of Increasing & Decreasing

Given a function y=f(x)y=f(x) and two points (x1,f(x1))(x_1,f(x_1)) and (x2,f(x2))(x_2,f(x_2)). Then,

A function is increasing when:
  • x1<x2x_1<x_2, then f(x1)≤f(x2)f(x_1)\leq{}f(x_2)
  • both xx and yy increase or decrease together

A function is decreasing when either:
  • x1<x2x_1<x_2, then f(x1)≥f(x2)f(x_1)\geq{}f(x_2)
  • xx increases and yy decreases OR;
  • xx decreases and yy increases.

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Example: Combining Families of Functions

Let f(x)=x(x4−4)f(x)=x(x^4-4).

Sketch a graph of the function and determine if the function is even or odd, points of maximums and minimums, and intervals of increasing and decreasing


Even or Odd

f(x)=x(x4−4)f(−x)=(−x)((−x)4−4)=−x(x4−4)f(−x)=−f(x)✓\begin{array}{rcl} f(x)&=&x(x^4-4)\\\\ f(-x)&=&(-x)((-x)^4-4)\\\\ &=&-x(x^4-4)\\\\ f(-x)&=&-f(x)&\color{red}\checkmark \end{array}

Therefore, the function is odd.


Maximums and Minimums

Max: ≈(−0.9,3)\approx(-0.9,3)

Min: ≈(0.9,−3)\approx(0.9,-3)


Intervals of Increasing and Decreasing

Inc: (−∞,−0.9) ∪ (0.9,∞)(-\infin,-0.9)~\cup~(0.9,\infin)

Dec: (−0.9,0.9)(-0.9,0.9)

Practice: Combining Families of Functions


Match the graph to the function.
A.
y=4xlog⁡3xy=\dfrac{4}{x}\log_{3}x
B.
y=∣x∣3xy=|x|{}^3\sqrt{x}
C.
y=x(2x)y=x(2^x)
D.
y=xsin⁡xy=x\sin{x}


Practice: Combining Families of Functions

A combined function is shown below.


What are the intervals of increasing and decreasing?

Practice: Combining Families of Functions

Is the function f(x)=4−x39x2+1f(x)=\dfrac{4-x^3}{9x^2+1} even, odd, or neither?

Extra Practice
Combining Families of Functions
Practice: Combining Families of Functions
Is the function f(x)=x2x2−6f(x)=\dfrac{x^2}{x^2-6} even, odd, or neither?
Combining Families of Functions
Practice: Combining Families of Functions
Is the function f(x)=xx2−1f(x)=x\sqrt{x^2-1} even, odd, or neither?
Combining Families of Functions
Practice: Combining Families of Functions
A combined function is shown below.
What are the intervals of increasing and decreasing?
Combining Families of Functions
Practice: Combining Families of Functions
A combined function is shown below.
What are the intervals of increasing and decreasing?
Combining Families of Functions
Practice: Combining Families of Functions
Match the graph to the function.
Combining Families of Functions
Practice: Combining Families of Functions
Match the graph to the function.