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A California county reports a backlog of one hundred applications.

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Will the backlog rise by ten applications each week, rise by ten percent each week, or rise and then fall after temporary staff arrive?

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Those are different mathematical stories.

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What evidence would let a policy analyst choose among them?

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This is study preparation only.

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It awards no score, course credit, or transfer credit.

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Use the audio to guide written practice, then check institutional rules before treating any outside course or exam as credit.

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A function is not just a formula.

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It is a rule that gives one output for each allowed input.

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You can describe it with words, a table, a graph, or an equation.

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Good algebra moves among all four forms.

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Start with a table.

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In week zero, a permit backlog is one hundred.

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In week one, it is one hundred ten.

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In week two, it is one hundred twenty.

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In week three, it is one hundred thirty.

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The first differences are ten, ten, and ten.

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A constant first difference signals a linear pattern.

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The spoken rule is: start at one hundred and add ten per week.

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The rate of change is ten applications per week.

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Now change the table.

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Week zero is one hundred.

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Week one is one hundred ten.

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Week two is one hundred twenty-one.

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Week three is about one hundred thirty-three.

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Here the ratios are constant because each value is one and one tenth times the previous value.

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This is exponential growth at ten percent per week.

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Linear growth adds the same amount.

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Exponential growth multiplies by the same factor.

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The two models agree after one week and separate after that.

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A convenient first data point is not enough to identify the pattern.

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Look across several periods, examine differences and ratios, and ask whether a staffing change or capacity limit altered the process.

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Function notation helps organize evaluation.

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If f of x equals x squared minus four x, then f of three means replace every x with three.

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Three squared is nine.

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Four times three is twelve.

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Nine minus twelve is negative three.

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Parentheses matter when substituting negative values.

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For f of negative two, negative two squared is four, while negative four times negative two contributes positive eight.

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The output is twelve.

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Domain restrictions come from both algebra and context.

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A denominator cannot equal zero because division by zero is undefined.

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A square root, when working with real numbers, cannot take a negative input.

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In a county workload model, time may be nonnegative whole weeks.

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A formula can accept a value that the public system cannot.

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Polynomial functions combine powers of a variable with numerical coefficients.

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A quadratic has degree two, meaning its highest power is two.

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Its graph is a parabola.

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It can decrease, turn, and increase, so a constant-rate story may fail.

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Factoring can reveal where a polynomial equals zero.

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Solve x squared minus five x plus six equals zero.

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Find two numbers that multiply to positive six and add to negative five.

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They are negative two and negative three.

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The expression factors as x minus two times x minus three.

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A product is zero when at least one factor is zero.

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Therefore x equals two or x equals three.

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Check: four minus ten plus six is zero.

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Nine minus fifteen plus six is also zero.

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The quadratic formula handles cases that do not factor easily.

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For today, recognize its purpose: it finds solutions to a quadratic equation using the coefficients.

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Also recognize that a negative value under its square root means there are no real-number solutions.

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Do not force a real answer when the algebra says none exists.

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A rational function is a ratio of polynomials.

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Consider one hundred divided by x, where x is the number of crews sharing one hundred inspection hours.

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With two crews, the model gives fifty hours per crew.

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With five, it gives twenty.

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With ten, it gives ten.

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But x cannot be zero.

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Near zero, the model behaves dramatically, which warns us that a rational relationship is not linear.

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Rational equations require checking for excluded values.

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Solve twelve divided by x equals three.

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Multiply both sides by x, provided x is not zero.

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Twelve equals three x.

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Divide by three, so x equals four.

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Check four in the original equation: twelve divided by four is three.

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The check confirms both the arithmetic and the domain.

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An inverse function reverses an input-output process when that reversal is unique.

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Suppose cost equals fifty plus twenty times the number of service batches.

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To recover batches from a known cost, subtract fifty, then divide by twenty.

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A cost of one hundred fifty gives one hundred after subtraction, then five batches after division.

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Not every function has an inverse over its full domain.

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Squaring sends both three and negative three to nine, so seeing nine does not reveal which input was used.

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Restricting the domain to nonnegative inputs makes the reversal unique.

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In public data, the same caution applies: one outcome may have several possible inputs.

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Exponential functions model constant percentage change.

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Begin with two hundred online requests and increase by five percent each month.

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Multiply by one point zero five.

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After one month, there are two hundred ten.

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After two months, multiply two hundred ten by one point zero five, giving two hundred twenty point five.

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A count would need sensible rounding, but keep the decimal during calculation to avoid compounding a rounding error.

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Exponential decay uses a factor between zero and one.

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A ten percent monthly reduction leaves ninety percent, so multiply by zero point nine.

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A backlog of five hundred becomes four hundred fifty after one month, then four hundred five after two months.

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Subtracting fifty each month would be linear decay and would tell a different story.

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A logarithm answers an exponent question.

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If two raised to the third power equals eight, then the base-two logarithm of eight is three.

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In policy forecasting, a logarithm can answer how many compounding periods it takes to cross a threshold.

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The base and the growth factor must match the model.

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Exponential equations can sometimes be solved by recognizing equal bases.

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Solve two raised to x equals thirty-two.

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Thirty-two is two raised to the fifth power, so x equals five.

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When equal bases are not obvious, logarithms isolate the exponent.

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A calculator gives an approximation, but the interpretation remains "the number of growth periods." Sequences describe ordered terms.

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An arithmetic sequence adds a constant.

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Starting at fifty and adding eight gives fifty, fifty-eight, sixty-six, and seventy-four.

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A geometric sequence multiplies by a constant.

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Starting at fifty and multiplying by two gives fifty, one hundred, two hundred, and four hundred.

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A series is the sum of sequence terms.

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Now apply the models locally.

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A California city sees one hundred code-enforcement cases, then one hundred ten, then one hundred twenty-one.

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The constant ratio suggests ten percent growth.

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Before projecting, ask whether reporting outreach caused a temporary jump, whether duplicate cases exist, and whether staff capacity caps recorded cases.

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Mathematics identifies a pattern in observed numbers.

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It does not certify the mechanism.

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Compare three forecasts for the next period.

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A linear rule based on adding ten predicts one hundred thirty-one.

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An exponential rule based on multiplying one hundred twenty-one by one point one predicts about one hundred thirty-three.

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A capacity rule might cap opened cases at one hundred twenty-five.

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Each answer can be calculated correctly while answering a different question.

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Three questions.

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First, how can a table distinguish linear from exponential change?

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Second, why must a rational-function solution be checked in the original equation?

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Third, what does a logarithm tell you in a growth model?

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First answer: constant differences suggest linear change, while constant nonzero ratios suggest exponential change.

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Second answer: algebraic steps may introduce an excluded value, especially a value that makes a denominator zero.

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Substitution confirms the candidate is permitted and true.

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Third answer: a logarithm gives the exponent, which often represents the number of compounding periods needed to reach a stated value.

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Artifact: complete a mixed review of at least twenty problems on function evaluation and domains, linear versus exponential tables, factoring and quadratics, rational expressions, inverses, exponentials, logarithms, sequences, and basic counting or probability.

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Earn at least eighty percent, correct every miss in writing, and label the model and units in each word problem.

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Source note: Original study audio based on public learning materials, without quoting source prose.

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See OpenStax, College Algebra 2e, Chapter 3, Functions;

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Chapter 4, Linear Functions;

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Chapter 5, Polynomial and Rational Functions;

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Chapter 6, Exponential and Logarithm Functions;

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and Chapter 9, Sequences, Probability, and Counting Theory.

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Access for free at openstax.org.

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OpenStax identifies College Algebra 2e as licensed under Creative Commons Attribution-NonCommercial-ShareAlike 4.0, or CC BY-NC-SA 4.0.

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Also consult the public Modern States College Algebra overview, especially algebraic operations, functions and their properties, equations and inequalities, and number systems and operations, plus Khan Academy public lessons on polynomials, rational expressions, exponential and logarithmic functions, sequences, and probability.

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Preparation only.

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No credit or transfer credit is awarded by this episode.