Precalculus Functions and Their Graphs Syllabus

Functions and Their Graphs syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Functions and Their Graphs in Precalculus, not a summary of it.

1. 1

   Foundations of Functions

   4 topics

   • Definition and Notation
   • Domain and Range
   • Piecewise-Defined Functions
   • Difference Quotient
2. 2

   Analyzing Graphs of Functions

   5 topics

   • Intercepts and Zeros
   • Increasing, Decreasing, and Constant Intervals
   • Relative Extrema
   • Symmetry and Even/Odd Functions
   • Average Rate of Change
3. 3

   Library of Parent Functions

   4 topics

   • Linear and Constant Functions
   • Power and Root Functions
   • Absolute Value and Greatest Integer Functions
   • Quadratic Function
4. 4

   Transformations of Functions

   4 topics

   • Vertical and Horizontal Shifts
   • Reflections
   • Stretches and Compressions
   • Combining Transformations
5. 5

   Combinations and Composition

   2 topics

   • Arithmetic Combinations of Functions
   • Composition of Functions
6. 6

   Inverse Functions

   3 topics

   • One-to-One Functions and Horizontal Line Test
   • Finding Inverse Functions Algebraically
   • Graphs of Inverse Functions

Functions and Their Graphs flashcards for Precalculus

19 of 51 cards from the Functions and Their Graphs deck — real questions with worked answers.

1. What is the definition of a function from a set $X$ to a set $Y$?

   A function is a rule that assigns to each element $x \in X$ (the domain) exactly one element $y \in Y$. No input may map to two different outputs.

2. In the function notation $y = f(x)$, what do $x$, $y$, and $f(x)$ represent?

   $x$ is the independent variable (input), $y$ is the dependent variable (output), and $f(x)$ denotes the value of the function $f$ at $x$.

3. State the Vertical Line Test and what it determines.

   A graph in the plane represents $y$ as a function of $x$ if and only if no vertical line intersects the graph more than once. It tests whether each input has a single output.

4. What is the domain of a function, and what is the range?

   The domain is the set of all permissible input values $x$; the range is the set of all resulting output values $f(x)$.

5. How do you find the domain of a function given by a formula?

   Include all real numbers except those that make a denominator zero or produce an even root of a negative number (and exclude values outside a defined restriction such as logs of nonpositive numbers).

6. Find the domain of $f(x) = \dfrac{1}{x-3}$.

   All real numbers except $x = 3$: $\{x : x \neq 3\}$, or $(-\infty, 3) \cup (3, \infty)$.

7. Find the domain of $g(x) = \sqrt{x-5}$.

   Require $x - 5 \geq 0$, so $x \geq 5$; the domain is $[5, \infty)$.

8. What is a piecewise-defined function?

   A function defined by different formulas (rules) on different parts of its domain, each rule applying to a specified interval of input values.

9. Evaluate $f(-2)$ and $f(3)$ for $f(x) = \begin{cases} x^{2} & x < 0 \\ 2x+1 & x \geq 0 \end{cases}$.

   Since $-2 < 0$, $f(-2) = (-2)^{2} = 4$. Since $3 \geq 0$, $f(3) = 2(3)+1 = 7$.

10. Write the formula for the difference quotient of a function $f$.

    $$\frac{f(x+h) - f(x)}{h}, \quad h \neq 0$$

11. Compute and simplify the difference quotient for $f(x) = x^{2}$.

    $$\frac{(x+h)^{2} - x^{2}}{h} = \frac{2xh + h^{2}}{h} = 2x + h$$

12. What does the difference quotient represent geometrically?

    The slope of the secant line through the points $(x, f(x))$ and $(x+h, f(x+h))$ on the graph of $f$.

13. How do you find the $y$-intercept of a function $y = f(x)$?

    Evaluate $f(0)$; the $y$-intercept is the point $(0, f(0))$, provided $0$ is in the domain.

14. What are the zeros (or $x$-intercepts) of a function $f$, and how are they found?

    The zeros are the $x$-values for which $f(x) = 0$; solve the equation $f(x)=0$. They correspond to the $x$-intercepts $(x, 0)$ of the graph.

15. Find the zeros of $f(x) = x^{2} - 5x + 6$.

    Factor: $x^{2}-5x+6 = (x-2)(x-3) = 0$, so the zeros are $x = 2$ and $x = 3$.

16. Define what it means for a function to be increasing on an interval.

    $f$ is increasing on an interval if for any $x_{1} < x_{2}$ in the interval, $f(x_{1}) < f(x_{2})$; the graph rises from left to right.

17. Define what it means for a function to be decreasing on an interval.

    $f$ is decreasing on an interval if for any $x_{1} < x_{2}$ in the interval, $f(x_{1}) > f(x_{2})$; the graph falls from left to right.

18. Define what it means for a function to be constant on an interval.

    $f$ is constant on an interval if $f(x_{1}) = f(x_{2})$ for all $x_{1}, x_{2}$ in the interval; the graph is a horizontal segment.

19. Define a relative (local) maximum of a function $f$.

    $f$ has a relative maximum at $c$ if there is an open interval containing $c$ on which $f(c) \geq f(x)$ for all $x$ in that interval. The value $f(c)$ is the relative maximum.

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