MA 161 Fall 2026 – Lesson 1 (08/24/26)

Original Notes

Page 1

MA 161 Fall 2026 – Lesson 1 (08/24/26)

Today:

  • Syllabus discussion
  • Introduction to the course
  • Review of functions

Announcements:

  • READ THE SYLLABUS
  • Get aquainted with course schedule
  • Enroll in MyLab Math HW System

Office Hours: MWF 1:30 PM – 2:30 PM

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What is a function?

Rule that assigns an output to each input

\[ \underset{\substack{\uparrow \\ \text{output}}}{y} = f(\underset{\substack{\uparrow \\ \text{input}}}{x}) \]

eg: \( y = f(x) = x^3 - 5x + 1 \)

\(\text{input} = 0 \rightsquigarrow \text{Output} = 0^3 - 5 \cdot 0 + 1 = 1\)

\(\text{input} = 1 \rightsquigarrow \text{Output} = 1^3 - 5 \cdot 1 + 1 = -3\)

\(\text{input} = k \rightsquigarrow \text{Output} = k^3 - 5k + 1\)

Page 3

Vertical line test

on a graph w input on Horizontal Axis
Ouput on vertical Axis

(1)
Graph illustrating the vertical line test for a function: A Cartesian coordinate system showing a smooth, strictly increasing blue curve passing from the third quadrant through the first quadrant. Four vertical red test lines are drawn across the graph, each intersecting the blue curve at exactly one point, demonstrating that every input corresponds to exactly one output.
Visual Description: Graph illustrating the vertical line test for a function: A Cartesian coordinate system showing a smooth, strictly increasing blue curve passing from the third quadrant through the first quadrant. Four vertical red test lines are drawn across the graph, each intersecting the blue curve at exactly one point, demonstrating that every input corresponds to exactly one output.

Graph of function

(2)
Graph illustrating the vertical line test for a non-function: A Cartesian coordinate system showing an S-shaped blue curve that loops back over itself horizontally. A vertical red test line passes through the curve and intersects it at three distinct vertical points, demonstrating that a single input produces multiple outputs, failing the vertical line test.
Visual Description: Graph illustrating the vertical line test for a non-function: A Cartesian coordinate system showing an S-shaped blue curve that loops back over itself horizontally. A vertical red test line passes through the curve and intersects it at three distinct vertical points, demonstrating that a single input produces multiple outputs, failing the vertical line test.

Not a graph of function.

Page 4

Domain and Range

Domain: all acceptable inputs

Range: All possible outputs

eg: \( f(x) = \sqrt{x+5} \)

  • \( x = 4 \rightarrow \text{output} = \sqrt{9} = 3 \)
  • \( x = 0 \rightarrow \text{output} = \sqrt{5} \)
  • \( x = -5 \rightarrow \text{output} = 0 \)
  • \( x = -5.01 \rightarrow \text{Not Acceptable input} \)

Domain: \( [-5, \infty) \)

Range: \( [0, \infty) \)

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eg: Find domain of \( f(x) = \sqrt{x-1} - \frac{1}{\sqrt{2-x}} \)

\( \sqrt{x-1} \rightarrow \) acceptable input is \( x \ge 1 \)

\( \sqrt{2-x} \rightarrow \) acceptable input is \( 2-x \ge 0 \) or \( x \le 2 \)

Want both \( \Rightarrow \) Domain is \( x \ge 1 \) and \( x \le 2 \)
\( [1, 2] \)

Not really \( \rightarrow \) Cannot have \( \sqrt{2-x} = 0 \Rightarrow x \neq 2 \)

\( \text{Domain} = [1, 2) \)

Page 6

Transformations

Shift:
Cartesian coordinate system illustrating graph transformations. The base curve \( y = f(x) \) is drawn in blue passing through the origin. A vertically shifted curve in purple, labeled 'Shift up' and \( y = h(x) \), is displaced upward along the y-axis by a distance of \( c \) units. A horizontally shifted curve in green, labeled 'Shift Right' and \( y = g(x) \), is displaced to the right by a distance of \( k \) units, indicated by dashed lines connecting the corresponding points at \( x - k \) and \( x \).
Visual Description: Cartesian coordinate system illustrating graph transformations. The base curve \( y = f(x) \) is drawn in blue passing through the origin. A vertically shifted curve in purple, labeled 'Shift up' and \( y = h(x) \), is displaced upward along the y-axis by a distance of \( c \) units. A horizontally shifted curve in green, labeled 'Shift Right' and \( y = g(x) \), is displaced to the right by a distance of \( k \) units, indicated by dashed lines connecting the corresponding points at \( x - k \) and \( x \).
Vertical Shift: \( c > 0 \)

\[ h(x) = f(x) + c \rightsquigarrow \text{up by } c \text{ units} \]

\[ h(x) = f(x) - c \rightsquigarrow \text{down by } c \text{ units} \]

Horizontal Shift: \( k > 0 \)

\[ g(x) = f(x - k) \rightsquigarrow \text{Right by } k \text{ units} \]

\[ g(x) = f(x + k) \rightsquigarrow \text{Left by } k \text{ units} \]

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Stretching!

Vertically!

\[ h(x) = c \, f(x) \]

\[ c > 1 \implies \text{stretching} \]

\[ 0 < c < 1 \implies \text{compressing} \]

Horizontally!

\[ g(x) = f(cx) \quad \text{or} \quad f\left(\frac{x}{c}\right) \]

For \( f(cx) \):

  • \( c > 1 \implies \text{compressing} \)
  • \( 0 < c < 1 \implies \text{stretch} \)

For \( f\left(\frac{x}{c}\right) \):

  • \( c > 1 \implies \text{stretch} \)
  • \( 0 < c < 1 \implies \text{compress} \)

Page 8

Reflection:

Graph illustrating function reflections on a Cartesian plane: the original curve y = f(x) is drawn in blue in the first quadrant; the curve y = g(x) in green represents the reflection of f(x) about the y-axis in the second quadrant; and the curve y = h(x) in purple in the fourth quadrant represents the reflection of f(x) about the x-axis. Dashed lines connect corresponding symmetric points at x and -x to illustrate horizontal and vertical reflections across the axes.
Visual Description: Graph illustrating function reflections on a Cartesian plane: the original curve y = f(x) is drawn in blue in the first quadrant; the curve y = g(x) in green represents the reflection of f(x) about the y-axis in the second quadrant; and the curve y = h(x) in purple in the fourth quadrant represents the reflection of f(x) about the x-axis. Dashed lines connect corresponding symmetric points at x and -x to illustrate horizontal and vertical reflections across the axes.

\( g(x) = f(-x) \) is reflection about y-axis

\( h(x) = -f(x) \) is reflection about x-axis.