Test of Mathematics for University Admission (TMUA) Functions, Sequences and Series Syllabus

Functions, Sequences and Series syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Functions, Sequences and Series in Test of Mathematics for University Admission (TMUA), not a summary of it.

1. 1

   Functions

   3 topics

   • Function Notation and Definitions
     • Domain and range
     • Mappings and well-defined functions
   • Composite Functions
     • Order of composition
     • Domain of a composite
   • Inverse Functions
     • Conditions for invertibility
     • Graph of an inverse as reflection in y=x
2. 2

   Exponentials and Logarithms

   4 topics

   • Exponential Functions
     • Properties and graphs of a^x
     • Exponential growth and decay
   • Laws of Logarithms
     • Product, quotient and power laws
     • Change of base
   • Solving Exponential Equations
     • Taking logarithms of both sides
     • Disguised quadratics in exponentials
   • Modelling with Logarithms
     • Linearising power and exponential laws
3. 3

   Sequences and Series

   4 topics

   • Arithmetic Sequences
     • nth term formula
     • Sum of an arithmetic series
   • Geometric Sequences
     • nth term and common ratio
     • Sum to n terms
     • Sum to infinity and convergence
   • Sigma Notation
     • Expanding and evaluating sums
   • Recurrence Relations
     • Generating terms iteratively
     • Periodic sequences
4. 4

   The Binomial Expansion

   3 topics

   • Binomial Theorem for Positive Integers
     • Pascal's triangle and coefficients
     • General term of the expansion
   • Binomial Coefficients
     • Factorial and nCr notation
   • Applications
     • Finding particular coefficients
     • Approximations from expansions

Functions, Sequences and Series flashcards for Test of Mathematics for University Admission (TMUA)

25 of 51 cards from the Functions, Sequences and Series deck — real questions with worked answers.

1. What does the notation $f: x \mapsto 2x+3$ mean, and how is it equivalently written?

   It defines a function $f$ that maps an input $x$ to the output $2x+3$. Equivalently written as $f(x) = 2x+3$. The arrow $\mapsto$ reads 'maps to'.

2. Define the domain and range of a function.

   The domain is the set of all permitted input values (values of $x$). The range is the set of all output values $f(x)$ produced as $x$ varies over the domain.

3. What is the formal condition for a relation to be a function?

   Each element of the domain maps to exactly one element of the range. No input may produce more than one output (the 'vertical line test').

4. Distinguish between a one-to-one and a many-to-one function.

   A one-to-one (injective) function maps distinct inputs to distinct outputs (each output comes from at most one input). A many-to-one function maps two or more different inputs to the same output, e.g. $f(x)=x^{2}$.

5. For the composite function $fg(x)$ (also written $f \circ g$), which function is applied first?

   $g$ is applied first, then $f$ is applied to the result: $fg(x) = f(g(x))$. Work from the inside out.

6. If $f(x)=x^{2}$ and $g(x)=x+1$, find $fg(x)$ and $gf(x)$.

   $fg(x) = f(x+1) = (x+1)^{2}$. $gf(x) = g(x^{2}) = x^{2}+1$. These are different, showing composition is generally not commutative.

7. What must be true about the range of $g$ for the composite $fg(x)=f(g(x))$ to be defined?

   The range of $g$ must lie within the domain of $f$, so that every output of $g$ is a valid input to $f$.

8. What condition must a function satisfy in order to have an inverse function?

   It must be one-to-one (injective) over its domain. A many-to-one function has no inverse unless its domain is restricted to make it one-to-one.

9. State the key relationships between a function $f$ and its inverse $f^{-1}$.

   $f f^{-1}(x) = x$ and $f^{-1} f(x) = x$. The domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$.

10. Describe the process for finding the inverse of $f(x) = \frac{3x-2}{x+1}$.

    Set $y = \frac{3x-2}{x+1}$, swap to solve for $x$ in terms of $y$: $y(x+1)=3x-2 \Rightarrow x(y-3) = -2-y \Rightarrow x = \frac{-2-y}{y-3}$. Then $f^{-1}(x) = \frac{x+2}{3-x}$.

11. What is the geometric relationship between the graph of $y=f(x)$ and the graph of $y=f^{-1}(x)$?

    They are reflections of each other in the line $y=x$.

12. Define an exponential function and state its general form.

    An exponential function has the form $f(x) = a^{x}$ where $a>0$, $a \neq 1$. The variable appears in the exponent. The special base is $e \approx 2.718$, giving $f(x)=e^{x}$.

13. State the key features of the graph of $y = a^{x}$ for $a>1$.

    It passes through $(0,1)$, is always positive, increases for all $x$, and has the $x$-axis ($y=0$) as a horizontal asymptote as $x \to -\infty$.

14. What is special about the gradient of the curve $y = e^{x}$?

    Its gradient at every point equals its $y$-value: $\frac{d}{dx}e^{x} = e^{x}$. The gradient at $(0,1)$ is exactly $1$.

15. Define $\log_{a} x$ in terms of exponentials.

    $\log_{a} x = y$ means exactly $a^{y} = x$. The logarithm is the inverse of the exponential: $y = a^{x} \Leftrightarrow x = \log_{a} y$, for $a>0$, $a\neq 1$, $x>0$.

16. State the three laws of logarithms (product, quotient, power).

    $\log_{a}(xy) = \log_{a} x + \log_{a} y$; $\log_{a}\!\left(\frac{x}{y}\right) = \log_{a} x - \log_{a} y$; $\log_{a}(x^{k}) = k\log_{a} x$.

17. What are the values of $\log_{a} 1$ and $\log_{a} a$?

    $\log_{a} 1 = 0$ (since $a^{0}=1$) and $\log_{a} a = 1$ (since $a^{1}=a$).

18. State the change of base formula for logarithms.

    $\log_{a} x = \frac{\log_{b} x}{\log_{b} a}$ for any valid base $b$. A common choice is $\log_{a} x = \frac{\ln x}{\ln a}$.

19. What is the natural logarithm, and how does it relate to $e^{x}$?

    The natural logarithm $\ln x = \log_{e} x$ is the logarithm to base $e$. It is the inverse of $e^{x}$: $\ln(e^{x}) = x$ and $e^{\ln x} = x$.

20. Solve $2^{x} = 10$ using logarithms.

    Take logs of both sides: $x \ln 2 = \ln 10$, so $x = \frac{\ln 10}{\ln 2} \approx 3.32$.

21. Describe how to solve a 'hidden quadratic' exponential equation such as $e^{2x} - 5e^{x} + 6 = 0$.

    Substitute $y = e^{x}$ to get $y^{2} - 5y + 6 = 0$, factorise to $(y-2)(y-3)=0$, so $y=2$ or $y=3$. Then $x = \ln 2$ or $x = \ln 3$.

22. How do you solve an equation where the unknown appears in exponents on both sides, e.g. $3^{x} = 5^{x-1}$?

    Take logs of both sides: $x\ln 3 = (x-1)\ln 5$. Expand and collect $x$ terms: $x(\ln 3 - \ln 5) = -\ln 5$, so $x = \frac{-\ln 5}{\ln 3 - \ln 5} = \frac{\ln 5}{\ln 5 - \ln 3}$.

23. A quantity follows $y = ab^{x}$. How can logarithms linearise this for graphing?

    Take logs: $\log y = \log a + x\log b$. Plotting $\log y$ against $x$ gives a straight line with gradient $\log b$ and intercept $\log a$.

24. A quantity follows a power law $y = ax^{n}$. How do you linearise it with logarithms?

    Take logs: $\log y = \log a + n\log x$. Plotting $\log y$ against $\log x$ gives a straight line of gradient $n$ and intercept $\log a$.

25. In the model $\log y = mx + c$ obtained from $y = ab^{x}$, how do you recover $a$ and $b$?

    Compare with $\log y = \log a + x\log b$: gradient $m = \log b$ so $b = 10^{m}$; intercept $c = \log a$ so $a = 10^{c}$ (using base-10 logs).

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