CSCA MATHEMATICS · MODULE 02 · LESSON 02.14

Mixed Practice: Functions, Sequences & Calculus

Learn to select an appropriate method when a question does not tell you which topic it tests. Review eight complete worked examples, error analysis and a final verification guide.

Method selection8 worked examplesAssessment in Tutor LMS quiz

01. How to Approach Mixed Problems

Choose the method from the question’s structure, not from the lesson heading.

  1. Identify the required result: domain, range, sum, slope, interval or optimal value.
  2. List restrictions from denominators, roots, logarithms or feasible dimensions.
  3. Choose the simplest applicable relationship.
  4. Check the result in the original problem.

For example, a quadratic minimum can be obtained by completing the square or by differentiation. The shorter correct method is preferable.

Study sequence

Review the method-selection guide, examine the eight worked examples, then complete the independent questions in the separate Tutor LMS quiz.

02. Method-Selection Guide

The first decision is often more important than the first calculation.

Question type First approach Essential check
Domain Combine algebraic restrictions Excluded inputs and endpoints
Range Analyze output values and extremes Are boundaries actually attained?
Monotonicity Read graph or derivative signs Domain breaks
Parity Compute \(f(-x)\) Symmetric domain
Graph identification Use shape, intercepts, asymptotes Match more than one feature
Exponential equation Rewrite using a common base Equate exponents only when justified
Logarithmic equation Check positive arguments first Verify candidates
Trigonometric value Exact values and identities Quadrant and angle units
Sequence term Common difference or ratio Term numbering
Finite sum Arithmetic or geometric sum Number of terms
Tangent equation Derivative plus curve point Slope and point
Optimization Form a one-variable objective Feasibility and extremum type

03. Domain and Range Together

WORKED EXAMPLE

Analyze a square-root rational function

Find the domain and range of:

\[f(x)=\sqrt{\frac{x-1}{x+2}}\]

1. Domain: Require a nonzero denominator and a nonnegative radicand.

\[x\neq-2,\qquad \frac{x-1}{x+2}\geq0\]

The sign changes at −2 and 1.

\[D=(-\infty,-2)\cup[1,\infty)\]

2. Range: Rewrite the radicand:

\[\frac{x-1}{x+2}=1-\frac{3}{x+2}\]

For \(x\) at least 1, the square root takes values from 0 up to but not including 1. For inputs below −2, it takes values strictly above 1. The value 1 would require \(x-1=x+2\), which is impossible.

\[R=[0,1)\cup(1,\infty)\]
Result: Domain: \((-\infty,-2)\cup[1,\infty)\). Range: \([0,1)\cup(1,\infty)\).

Key lesson

Approaching a value is not the same as reaching it.

04. Parity and Monotonicity

WORKED EXAMPLE

Analyze the cubic function

Determine the parity and increasing/decreasing intervals of:

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

The domain is all real numbers. Substitution gives:

\[f(-x)=-x^3+3x=-f(x)\]

Therefore the function is odd. Differentiate and factor:

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

Critical inputs are −1 and 1. Test the derivative on the intervals they form.

Interval Derivative sign Function behavior
\((-\infty,-1)\) Positive Increasing
\((-1,1)\) Negative Decreasing
\((1,\infty)\) Positive Increasing

Key lesson

Parity describes symmetry; monotonicity describes how outputs change as inputs increase.

05. A Logarithmic Equation

WORKED EXAMPLE

Solve and reject invalid roots

Solve:

\[\log_2(x-1)+\log_2(x-3)=3\]

1. Restrictions: Both arguments must be positive, so \(x>3\).

2. Combine and convert:

\[\log_2\left((x-1)(x-3)\right)=3\]
\[(x-1)(x-3)=8\]
\[x^2-4x-5=(x-5)(x+1)=0\]

The candidates are 5 and −1; only 5 meets the original argument restrictions.

\[\log_2 4+\log_2 2=2+1=3\]
Result: \(x=5\)

06. Recovering an Arithmetic Sequence

WORKED EXAMPLE

Use two known terms and find the sum

Given \(a_3=8\) and \(a_7=20\), find the general term and first ten-term sum.

\[d=\frac{20-8}{7-3}=3\]
\[a_1=a_3-2d=8-6=2\]
\[a_n=2+3(n-1)=3n-1\]
\[a_{10}=29,\qquad S_{10}=\frac{10}{2}(2+29)=155\]
Result: \(a_n=3n-1\); \(S_{10}=155\)

Index reminder

From the third to the seventh term there are four steps, not five.

07. A Tangent with a Specified Direction

WORKED EXAMPLE

Find a tangent parallel to a line

Find the tangent to \(f(x)=x^2-4x+1\) that is parallel to \(y=2x+7\).

Parallel lines have equal slopes. Differentiate and solve:

\[f'(x)=2x-4=2\quad\Rightarrow\quad x=3\]

The point is:

\[f(3)=9-12+1=-2\]

Use point–slope form:

\[y+2=2(x-3)\]
\[y=2x-8\]
Result: Point \((3,-2)\); tangent \(y=2x-8\).

08. Choosing the Right Optimization Model

WORKED EXAMPLE

Maximum area with three fenced sides

A farmer has 40 m of fencing and a straight wall forms the fourth side of a rectangle. Find the maximum area.

Let \(x\) be each perpendicular side and \(y\) the side parallel to the wall:

\[2x+y=40\quad\Rightarrow\quad y=40-2x\]
\[A(x)=x(40-2x)=40x-2x^2\]

The feasible interval is \(0<x<20\). Differentiate:

\[A'(x)=40-4x=0\quad\Rightarrow\quad x=10\]

The derivative changes from positive to negative at 10, so this is the absolute maximum. Then:

\[y=40-2(10)=20\]
\[A(10)=10(20)=200\]
Result: Dimensions 10 m × 20 m; maximum area 200 m².

Key modeling distinction

A wall replaces one side of fencing; do not impose the four-sided perimeter equation.

09. Error Analysis: Recognize and Repair

These are explained misconceptions, not independent quiz questions.

A. Missing denominator restriction

\[f(x)=\frac{\sqrt{x+1}}{x-3}\]

Checking the square root alone misses \(x=3\).

Correct relationship:

\[D=[-1,3)\cup(3,\infty)\]

B. Misusing the logarithm product rule

\[\log_2(x+4)\neq\log_2 x+\log_2 4\]

The rule separates products, not sums. For example, at \(x=4\), the left is 3 and the proposed right is 4.

Correct relationship:

\[\log_a(uv)=\log_a u+\log_a v\]

C. Off-by-one geometric exponent

\[a_n\neq a_1r^n\quad\text{in general}\]

Indexing from the first term requires zero ratio multiplications for \(n=1\).

Correct relationship:

\[a_n=a_1r^{n-1}\]

D. Tangent equation without a point

\[f'(2)=5\]

Knowing the slope alone does not determine the vertical position of a line.

Correct relationship:

\[y-f(2)=5(x-2)\]

E. Assuming a stationary input is a maximum

\[A'(x)=0\]

A stationary input may be a minimum, maximum, neither, or infeasible. Check the domain and sign change; on closed intervals compare endpoints.

10. Verify a Solution Efficiently

Problem type Quick verification
Domain Test boundaries and denominator zeros
Range Check endpoints and whether outputs are attained
Parity Check domain symmetry and compare \(f(-x)\)
Equations Substitute candidates into the original
Trigonometry Check units, signs and magnitude
Sequences Verify term indices and common differences/ratios
Derivatives Check coefficients, exponents and signs
Tangent Verify both point and slope
Optimization Check constraints, sign test and endpoints where applicable
Motion Check direction changes before computing total distance

Sanity check

In the arithmetic sum \(7+10+13+\cdots+34\), there are ten terms. Its sum must be between 70 and 340; this rules out obviously impossible answers before exact calculation.

11. Review Your Quiz Performance

Use the separate Tutor LMS quiz results to decide what to revise.

Quiz question numbers Revision focus
1–2 Domain, range and endpoints
3–4 Parity, monotonicity and graphs
5–6 Exponential and logarithmic equations
7–8 Trigonometry and quadrants
9–12 Sequences and finite sums
13–15, 18 Derivatives, tangents and stationary points
16 Optimization
17 Distance and displacement

For a missed or guessed question, record your original method, the precise error, and one check that would prevent it. Re-solve the problem without the answer key.

12. Lesson Review

  1. Select a method from the mathematical structure of the question.
  2. Combine all restrictions when finding a domain.
  3. Distinguish attained outputs from limiting outputs.
  4. Analyze symmetry and monotonicity separately.
  5. Verify exponential and logarithmic equation candidates.
  6. Use correct indexing for sequence terms and sums.
  7. Find a tangent from its slope and curve point.
  8. Check feasibility when optimizing, and distinguish distance from displacement.

Next step

Complete the independent mixed-practice questions in the separate quiz. The lesson intentionally contains no standalone questions or answer key.
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