MAT151 Final Exam Study Guide: College Algebra Essentials

Functions & Function Notation

Definition and Identification

A function is a relation where each input (x-value) has exactly one output (y-value). Function notation, such as f(x), represents the output for a given input x.

• Ordered Pair/Table Test: If an x-value repeats with different y-values, the relation is not a function.

• Vertical Line Test: If any vertical line intersects the graph more than once, it is not a function.

• Function Notation: f(3) means "the output when the input is 3." It is not multiplication.

• Inverse Notation: If f is a function, f-1 swaps input and output.

Example: If f(2) = 9, then f-1(9) = 2.

Domain, Range, Interval Notation, Piecewise Functions

Domain and Range

The domain is the set of allowed inputs (x-values), and the range is the set of possible outputs (y-values).

• Interval Notation: Brackets [ ] indicate inclusion; parentheses ( ) indicate exclusion.

• Conversion: -1 ≤ x < 3 becomes [−1, 3).

• Union: If x ≠ 4, domain is (−∞, 4) ∪ (4, ∞).

Domain Restrictions Checklist

• Fractions: Denominator ≠ 0

• Square roots: Inside ≥ 0

• 1 over a root: Inside > 0

• Logs: Inside > 0

Piecewise Functions

• Identify which interval x belongs to, then plug in.

• Graph with open/closed circles to indicate inclusion/exclusion.

Average Rate of Change & Graph Behavior

Average Rate of Change (AROC)

AROC measures the change in output per unit change in input, equivalent to the slope of the secant line.

• Formula: $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$

• AROC > 0: Increasing; AROC < 0: Decreasing; AROC = 0: No net change.

Graph Behavior

• Increasing: Graph rises left to right.

• Decreasing: Graph falls left to right.

• Relative Extrema: Local peaks (maxima) or valleys (minima).

• Concavity: Concave up (cup shape), concave down (cap shape).

• Inflection Point: Where concavity changes.

Function Operations, Composition, Decomposition

Composition of Functions

Composition applies one function to the result of another: (f ∘ g)(x) = f(g(x)).

• To evaluate f(g(3)): First compute g(3), then plug into f.

• Domain of Composition: x must be allowed in g, and g(x) must be allowed in f.

Decomposition

• Reverse-engineer a composite function into its inside and outside parts.

• Example: If h(x) = (2x + 3)^3, then inside: g(x) = 2x + 3, outside: f(u) = u^3.

Transformations & Even/Odd/Neither Functions

Transformations

Transformations modify parent functions by shifting, reflecting, or stretching/compressing.

• f(x) + k: Up k units

• f(x) − k: Down k units

• f(x − h): Right h units

• f(x + h): Left h units

• −f(x): Reflect over x-axis

• f(−x): Reflect over y-axis

• a f(x): Vertical stretch/compress by |a| (reflect if a < 0)

Even/Odd/Neither

• Even: f(−x) = f(x) (y-axis symmetry)

• Odd: f(−x) = −f(x) (origin symmetry)

Inverses & Linear Functions

Inverses

An inverse function swaps inputs and outputs. The graph of a function and its inverse are reflections across y = x.

• One-to-One Test: Horizontal line test; if any horizontal line hits more than once, not one-to-one.

• Finding an Inverse:

  1. Write y = f(x)

  2. Swap x and y

  3. Solve for y

  4. Rename as f-1(x)

• Verification: f(g(x)) = x and g(f(x)) = x

Linear Functions

• Slope-Intercept Form: y = mx + b (m = slope, b = y-intercept)

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Slope from Two Points: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Regression: Best-fit line; interpret slope and intercept in context.

Absolute Value, Power, and Polynomial Functions

Absolute Value

Absolute value measures distance from zero; outputs are never negative.

• Equation: |A| = b with b ≥ 0 gives two cases: A = b or A = −b

• Inequality:

  • |A| < b ⇒ −b < A < b

  • |A| ≤ b ⇒ −b ≤ A ≤ b

  • |A| > b ⇒ A < −b or A > b

  • |A| ≥ b ⇒ A ≤ −b or A ≥ b

Power Functions

• Must be exactly f(x) = kxp (one term, no extra +/− terms).

Polynomial Vocabulary

• Term: Each part separated by + or −

• Coefficient: Number multiplying a variable

• Constant Term: Term without a variable

• Degree: Highest power of x

• Leading Term/Coefficient: Term with highest degree

End Behavior

• Leading term a xn decides tails:

• Even n: Tails same direction

• Odd n: Tails opposite direction

• Sign of a: Up/down

Quadratic Functions

Forms and Features

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex Form: $f(x) = a(x - h)^2 + k$ (vertex at (h, k))

• Axis of Symmetry: $x = \frac{-b}{2a}$

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Discriminant: $b^2 - 4ac$ tells number of real solutions

Applications

• Vertex gives maximum or minimum value, depending on sign of a.

Graphs of Polynomial Functions, Inequalities, Difference Quotient

Intercepts and Multiplicity

• y-intercept: f(0)

• x-intercepts: Solve f(x) = 0

• Multiplicity: If factor is (x − h)p:

• p odd: Crosses the axis

• p even: Bounces/touches the axis

Polynomial Inequalities

• Move all terms to one side

• Factor completely

• Find critical values (zeros and undefined points)

• Test sign in each interval

• Select intervals matching the inequality

Difference Quotient

• Formula: $\frac{f(x + h) - f(x)}{h}$

• Use parentheses when subtracting f(x)

Rational Functions

Definition and Features

• Rational Function: $f(x) = \frac{P(x)}{Q(x)}$, Q(x) ≠ 0

• Factor numerator and denominator first

• If a factor cancels: Hole (removable discontinuity)

• If denominator = 0 after simplifying: Vertical asymptote

Horizontal Asymptotes (Degree Rule)

deg(P) = m	deg(Q) = n	Horizontal Asymptote
m < n		y = 0
m = n		y = (lead coeff P)/(lead coeff Q)
m > n		No horizontal asymptote (may have slant)

Solving Rational Equations/Inequalities

• Multiply both sides by LCD, solve, reject invalid solutions

• For inequalities: Factor, find critical numbers, test signs

Exponential Functions

Core Form and Meaning

• Exponential Function: $f(x) = a \cdot b^x + k$

• Initial value: f(0) = a + k

• Asymptote: y = k

• Growth if b > 1; decay if 0 < b < 1

Solving Exponential Equations

• Match bases, factor, or use logarithms

• If $b^x = c$, then $x = \frac{\ln c}{\ln b}$

Applications

• Discrete percent change: $A(t) = A_0(1 \pm r)^t$

• Compound interest: $A = P(1 + \frac{r}{n})^{nt}$

• Continuous growth/decay: $A = Pe^{rt}$

Logarithms, Models, Systems of Equations

Logarithms

• Definition: If $b^x = y$, then $\log_b(y) = x$

• Product Rule: $\log_b(MN) = \log_b M + \log_b N$

• Quotient Rule: $\log_b(M/N) = \log_b M - \log_b N$

• Power Rule: $\log_b(M^p) = p \log_b M$

• Domain Rule: Log input must be positive (inside > 0)

Growth Rate Conversion

• Continuous to annual: $r = e^k - 1$

• Annual to continuous: $k = \ln(1 + r)$

Systems of Equations & Matrices

• Linear Systems: Solve by substitution or elimination

• Non-linear Systems: Substitution often leads to quadratic; solutions are intersection points

• Matrices: Row-reduce (swap, scale, add rows)

• Systems of Inequalities: Shade regions; solution is overlap

Toolkit Parent Functions

Function Name	Equation
Constant	$f(x) = c$
Identity	$f(x) = x$
Absolute Value	$f(x) = |x|$
Quadratic	$f(x) = x^2$
Cubic	$f(x) = x^3$
Square Root	$f(x) = \sqrt{x}$
Reciprocal	$f(x) = \frac{1}{x}$
Reciprocal Squared	$f(x) = \frac{1}{x^2}$

Final Exam Top Traps

• Forgetting domain restrictions (logs, rationals, even roots)

• Inside is backwards on transformations

• Inverse requires one-to-one (horizontal line test)

• Rational: Canceling factor means hole, not vertical asymptote

• Absolute value: Always two cases when |A| = b and b ≥ 0

• Difference quotient: Use parentheses or signs may flip

Summary Table: Key Concepts and Formulas

Concept	Formula/Rule
Average Rate of Change	$\frac{f(x_2) - f(x_1)}{x_2 - x_1}$
Composition	$(f \circ g)(x) = f(g(x))$
Inverse Steps	Write y = f(x), swap x/y, solve for y, rename
Slope	$m = \frac{y_2 - y_1}{x_2 - x_1}$
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Difference Quotient	$\frac{f(x + h) - f(x)}{h}$
Exponential Equation	$b^x = c \Rightarrow x = \frac{\ln c}{\ln b}$
Logarithm Definition	$b^x = y \Leftrightarrow \log_b(y) = x$

Additional info: This guide covers all major topics from college algebra, including functions, equations, graphs, polynomials, rational functions, exponentials, logarithms, systems, and matrices. It is structured for exam preparation, with concise explanations, formulas, and examples.