Analyzing Functions

Inequality: x > -3

  • <, >, ≤, ≥

Set Builder: {xlx > -3}

  • {xl 2 ≤ x < 4} —> and

  • {xl 2 > x or x ≥ 4} —> or

Interval notation: ( -3, ∞ )

  • ( ) open circle

  • [ ] closed circle

  • ALWAYS order from least to greatest

  • (-3,4) U [7, 12) —> or

  • (-3,4) —> and

Infinity: can be +∞ or -∞

* ∞ cannot have [ ] as it is infinite

All real numbers: R

  • Set builder: {xlx ∈ R}

  • Inequality: -∞ < x < +∞

  • Interval notation: (-∞, ∞)

No solution: ∅

Graph of an inequality:

Graphing Inequalities in One VariableRelation: a set of ordered pairs

  • EX: {(4,5), (6,5), (7,-2), (8,6)}

*This set only works for discrete functions

Domain: a set of all x coordinates, ordered from least to greatest

  • EX: {4, 6, 7, 8}

Range: a set of all y coordinates, ordered from least to greatest

  • EX: {-2, 5, -6}

Function: There is one x for one y, which will pass the vertical line test

Discrete Function: A function but the values are not connected to each other

Continuous Function: a line that continues infinitely without breaks

y-Intercept: (0,y)

x-intercept: (x,0)

Increasing: Wherever the line is increasing on the graph

  • only use x values in your answer

Decreasing: wherever the line is decreasing on the graph

  • only use x values in your answer

End behavior:

  • As x —> ∞ , F(x) —> __

  • As x —> -∞ , F(x) —> __

Max: the largest y value of the line

Min: the smallest y value on the line

Pos: line points up

Neg: line points down

Horizontal shift (left/right) : g(x)= f(x-h)

  • -h is the opposite

    • so g(x)= f(x-7) —> +7

Vertical shift (up/down) : g(x)= f(x)+ k

Combined transformation: g(x)= f(x-h) +k

Vertical Reflections (reflects over the x-axis): g(x)= - f(x)

Horizontal Reflections (reflects over the y-axis): g(x)= f( -x)

Vertical Stretch: g(x)= af(x)

  • a>1

Vertical Compression: g(x)= af(x)

  • 0<a>1

Horizontal Stretch: g(x)= f(bx)

  • 0<b<1

  • b is the opposite

    • so 1/3 —> 3 ,and 4 —> ¼

Horizontal Compression: g(x)= f(bx)

  • b>1