Lec8 Review for Exam1 M2001 Odd Even Function, Transformation, Domain, Range, Inverse Function

Overview

Shahera Jannah reviews the M2001 Exam 1 topics, including line equations and linear models, function tests, odd/even symmetry, graph transformations, domains and ranges, and finding inverse functions. The review emphasizes showing algebraic work, identifying domain restrictions from radicals and denominators, and applying transformations in order; the exam is described as having no calculator section.

Key takeaways

For the score data (130, 470) and (142, 386), the linear model is s = −7m + 1,380; the negative slope represents a 7-point decrease per additional study minute across those two observations.
A $10 price increase paired with a 5-package sales decrease produces N(p) = −(1/2)p + 250, with price p as input and package count N(p) as output.
A graph passes the function test only if every vertical line intersects it at most once; a circle and a vertical line fail, while a standard parabola passes.
Radical and denominator restrictions must be combined: √x with denominator 1 − √x has domain [0, 1) ∪ (1, ∞), while 1/√(4 − x²) has domain (−2, 2).
In f(x) = −g(x + 1) − 2, the negative outside reflects across the x-axis, x + 1 shifts left 1, and −2 shifts down 2; apply the reflection before translating.
To find the inverse of f(x) = (2 − 3x)/(4x + 5), swap x and y and solve, yielding f⁻¹(x) = (2 − 5x)/(4x + 3).

Chapters

0:00 Exam Review Scope: Line Equations and Slope Relationships

The review begins with line equations through two points, through one point with a given slope, and for parallel or perpendicular lines.
Parallel lines have the same slope; perpendicular slopes are negative reciprocals.
For standard form ax + by = c, the slope shortcut is −a/b; the examples include slopes −7/6 and 5/8.

4:00 Linear Score Model from Two Study-Time Data Points

The study-score data are (130 minutes, 470 points) and (142 minutes, 386 points).
The slope is (386 − 470)/(142 − 130) = −7 points per minute.
Using b = y − mx gives an intercept of 1,380, so the model is s = −7m + 1,380.
The instructor says the exam has no calculator section and stresses showing work for both slope and intercept.

10:00 Package Sales Model: Price as Input, Packages as Output

At a price of $100, sales are 200 packages; raising the price by $10 reduces sales by 5, giving the second point (110, 195).
The slope is −5/10 = −1/2 package per dollar, and the intercept is 250.
The requested function is N(p) = −(1/2)p + 250; use N(p) and p rather than generic y and x in the final answer.
The example highlights interpreting which quantity is the input and which is the function output.

17:00 Recognizing Functions with Graphs and Equations

The vertical line test checks graphs: a circle fails because some vertical lines intersect it twice, while a downward-opening parabola passes.
A vertical line fails the test and therefore is not a function of x.
For equations, the key idea is whether each input x determines only one output y; the lesson also presents a shortcut based on powers of y.

21:00 Even Polynomial Patterns and Constant Shifts

The lesson identifies x⁴, x², and their sums as even functions because replacing x with −x leaves their values unchanged.
Adding or subtracting a constant preserves evenness; for example, x⁴ − 1 remains even.
The graph-based explanation is symmetry about the y-axis, which remains after a vertical shift.

25:00 Testing Odd and Even Functions by Substitution

For f(x) = x⁴ − 2x + 1, replacing x with −x changes the middle-term sign, so the function is neither even nor odd.
The algebraic tests are f(−x) = f(x) for evenness and f(−x) = −f(x) for oddness.
The lesson demonstrates evaluating at 1 and −1 as an informal check, but matching values at a single pair does not by itself prove a function is even or odd.

35:00 Why Adding a Constant Usually Breaks Odd Symmetry

An odd function such as x³ has origin symmetry, while adding a nonzero constant shifts the graph vertically and generally removes that symmetry.
The example x³ + 3 is described as neither odd nor even after its vertical shift.
By contrast, a vertical shift of an even function preserves its y-axis symmetry.

39:00 Parity Checks for Rational and Mixed Expressions

The expression 1/x² + 7 is even because replacing x with −x leaves the value unchanged.
For a mixed expression such as x³/(x² + 1), the lesson compares values at 1 and −1 to illustrate odd behavior.
The example (x²)/(1 − x³) is treated as neither even nor odd; its domain also excludes x = 1.
The instructor advises checking mixed odd/even combinations instead of classifying them by appearance alone.

48:00 Fractional Powers and Symmetry Types

A function involving x^(2/3) is examined at 1 and −1; both inputs produce the same output, illustrating even behavior.
Even functions are symmetric about the y-axis; odd functions have symmetry about the origin.
The review distinguishes x-axis, y-axis, and origin symmetry, though the reliable algebraic tests are the identities f(−x) = f(x) and f(−x) = −f(x).

55:00 Reading Domain and Range from Graphs

Domain records x-values and range records y-values.
Use parentheses for excluded endpoints such as holes and brackets for included endpoints such as closed circles.
If a graph extends without omissions across all x-values or y-values, the corresponding interval can be (−∞, ∞).

1:00:00 Domain Rules for Radicals and Rational Functions

For a square root in the numerator or elsewhere outside a denominator, require the radicand to be greater than or equal to zero.
For a denominator, require the entire denominator to be nonzero and exclude its zeros from the domain.
When several factors determine a radical inequality, find critical values and use a number line to test intervals.

1:06:00 Domain Example: Radical in a Denominator

For the example with √x and denominator 1 − √x, the radical requires x ≥ 0.
The denominator vanishes at x = 1, so that value must be excluded.
Combining both restrictions gives the domain [0, 1) ∪ (1, ∞).

1:10:00 Domain Example: Square Root of 4 − x² in the Denominator

For a denominator containing √(4 − x²), the radicand must be strictly positive because a zero denominator is not allowed.
Solving 4 − x² > 0 gives −2 < x < 2.
The domain is (−2, 2); unlike a numerator radical, the endpoints cannot be included.

1:14:00 Transformation Rules for Reflections and Shifts

A negative sign outside a function, as in −f(x), reflects its graph across the x-axis.
A negative input, as in f(−x), reflects across the y-axis.
An inside shift f(x + 2) moves left 2, while an outside shift f(x) + 1 moves up 1.

1:20:00 Transforming a Graph with Reflection and Translation

For −f(x + 1) − 2, first reflect the original graph across the x-axis, then move it left 1 and down 2.
The worked graph uses key points, including the origin and points at x = −1 and x = 1, to track the transformation.
The reflected graph is translated only after the reflection is completed.

1:23:00 Applying the Same Transformation to a Second Graph

A second example repeats the reflection across the x-axis, followed by a 1-unit shift left and a 2-unit shift down.
The graph is reconstructed by reflecting its bends and turning points, then translating each point.
The example distinguishes the original graph, the reflected graph, and the final translated graph.

1:27:00 Finding the Inverse of a Rational Function

For f(x) = (2 − 3x)/(4x + 5), the procedure is to write y, swap x and y, and solve for y.
After cross-multiplying and collecting y terms, the inverse is f⁻¹(x) = (2 − 5x)/(4x + 3).
The instructor notes that graphing is not required for this algebraic inverse problem.

1:30:35 Closing Exam Advice and a Quick Parity Check

The closing reminders are to arrive on time, study strategically, and email questions if needed.
A student asks whether substituting 1 and −1 can simplify odd/even checks; the instructor says it can be used as a shortcut.
For a conclusive classification, compare the full expressions f(−x), f(x), and −f(x), since one pair of numerical evaluations may not establish a general identity.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Shahera Jannah's Math World! 😁😎.