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PAGE 1

1.1 + 1.2 Review of Functions

(there are 3 parts to the first HW)

Function:

\[ y = f(x) \]

Where y is the output and x is the input.

For example,

  • \[ f(x) = 2x^2 - x + 2 \]
  • \[ f(1) = 2(1)^2 - (1) + 2 = 3 \]
  • \[ f(0) = 2(0)^2 - 0 + 2 = 2 \]
  • \[ f(x-3) = 2(x-3)^2 - (x-3) + 2 \]

Graphs of functions must pass the vertical line test

Each vertical line must intersect the graph at most once.

Coordinate graph with a green curve and multiple vertical red lines, each intersecting the curve exactly once.
Coordinate graph of a sideways parabola where a vertical red line intersects the green curve at two points.

Crosses more than once. NOT graph of a function.

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Domain: all acceptable input values

Acceptable:

\[ f(x) = \frac{3}{x-2} \]\[ f(1) = \frac{3}{1-2} = \frac{3}{-1} = -3 \quad \text{number out} \]

Unacceptable:

\[ f(x) = \frac{3}{x-2} \]\[ f(2) = \frac{3}{2-2} = \frac{3}{0} \quad \text{undefined NOT a number} \]
  • no division by zero
  • no even roots of negative number
  • etc

Range: all possible output values

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Example: Domain and Range of a Square Root Function

Function Definition

\[ f(x) = \sqrt{x-5} \]

Finding the Domain

Domain: no even root of negative numbers.

So, \[ x - 5 \geq 0 \]

Solving for \( x \): \[ x \geq 5 \quad \text{or} \quad [5, \infty) \quad \text{or} \]

A number line with a solid dot at 5 and a bold blue line extending to the right.

Finding the Range

Range: the smallest possible value of \( x - 5 \) within the domain is \( 0 \) (when \( x = 5 \)).

So, the smallest \( \sqrt{x-5} \) can be is \( 0 \).

There is no maximum to \( x - 5 \) within the domain.

So there is no max to \( \sqrt{x-5} \) either.

Therefore, the range of \( \sqrt{x-5} \) is:

\[ [0, \infty) \]
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Composition of Functions

Composition of functions \( \rightarrow \) using a function as input of a function.

Given Functions

\[ f(x) = x^2 \]
\[ g(x) = \frac{1}{x-1} \]

Calculating \( f(g(x)) \)

\[ f(g(x)) = (f \circ g)(x) \]\[ = f\left(\frac{1}{x-1}\right) = \left(\frac{1}{x-1}\right)^2 \]
Change all \( x \) in \( f(x) \) to \( \frac{1}{x-1} \)

Calculating \( g(f(x)) \)

\[ g(f(x)) = (g \circ f)(x) \]\[ = g(x^2) = \frac{1}{x^2 - 1} \]
Change all \( x \) in \( g(x) \) to \( x^2 \)
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Symmetry

If \( f(-x) = f(x) \) then \( f(x) \) is even and its graph has y-axis symmetry.

Example:

\( f(x) = x^2 \)

\( f(-x) = (-x)^2 = x^2 = f(x) \)

Graph of y=x^2 showing symmetry across the y-axis with points (-1, 1) and (1, 1) labeled.

If \( f(-x) = -f(x) \) then \( f(x) \) is odd and its graph has origin symmetry.

Example:

\( f(x) = x^3 \)

\( f(-x) = (-x)^3 = -x^3 = -(x^3) = -f(x) \)

Graph of y=x^3 showing origin symmetry with points (-1, -1) and (1, 1) indicated.
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Graph Transformation

Vertical Shift

\[ y = f(x) + c \]

Shift vertically by \( c \)

Horizontal Shift

\[ y = f(x + b) \]

Shifts horizontally by \( b \)

  • If \( b > 0 \) → shift LEFT
  • If \( b < 0 \) → shift RIGHT
Coordinate graph showing transformations of f(x) including vertical shift f(x)+2 and horizontal shift f(x-2).

Vertical Stretch or Compression

\[ y = a f(x) \]
  • Stretch if \( a > 1 \)
  • Compression if \( 0 < a < 1 \)

Horizontal Stretch or Compression

\[ y = f(bx) \]
  • Stretch if \( 0 < b < 1 \)
  • Compress if \( b > 1 \)

Reflections

\( y = -f(x) \)Vertical reflection
\( y = f(-x) \)Horizontal reflection