Understanding Functions, Domain, and Range - Algebra 1

1. Identify the output when $f(x)=x^2-1$ and the input is $x=3$.

   $8$. Substitute $x=3$: $f(3)=3^2-1=9-1=8$.

2. What does it mean if $f(3)=10$?

   When $x=3$, the output is $10$. Function notation shows the input-output relationship.

3. What is the output variable called when the function is written as $y=f(x)$?

   $y$ (the dependent variable). The output depends on the input value chosen.

4. What does the vertical line test require for a graph to represent a function?

   Every vertical line intersects the graph at most once. This ensures no input has multiple outputs.

5. What is the correct reading of $f(7)=2$ in words?

   The function value at $7$ is $2$. This describes evaluating the function at input $7$.

6. What is the key rule that distinguishes a function from a general relation?

   No input is paired with more than one output. This is the fundamental property that defines a function.

7. Identify the domain of the relation $\{(1,2),(2,5),(3,5)\}$.

   $\{1,2,3\}$. Domain contains all first coordinates of ordered pairs.

8. What is the domain of $f(x)=\frac{1}{x}$?

   All real numbers except $0$. Division by zero is undefined, excluding $x=0$.

9. Evaluate $f(-1)$ for $f(x)=x^2+2$.

   $3$. Substitute $x=-1$: $f(-1)=(-1)^2+2=1+2=3$.

10. What must be true about the $y$-values for a single $x$ on the graph of a function?

    There can be only one $y$-value for that $x$. Functions cannot have multiple $y$-values for one $x$-value.

11. What word names the set of all allowed inputs of a function?

    Domain. Domain is the set of all valid input values.

12. What is the range of $f(x)=x^2$ over the real numbers?

    All real numbers $y$ such that $y\ge 0$. Squaring any real number produces a non-negative result.

13. Identify whether the equation $x=3$ represents $y$ as a function of $x$.

    Not a function (one $x$ value corresponds to many $y$ values). Vertical lines fail the vertical line test for functions.

14. Identify the input-output pair for $f(x)=x-6$ when the input is $x=6$.

    $(6,0)$. Calculate $f(6)=6-6=0$, giving pair $(6,0)$.

15. Identify whether the equation $y=3$ represents $y$ as a function of $x$.

    Function (each $x$ gives the single output $3$). Horizontal lines pass the vertical line test.

16. What is the output value called when the input is $x$ and the function is $f$?

    $f(x)$. This is the standard notation for function output values.

17. Identify whether $\{(0,2),(1,2),(2,2)\}$ is a function.

    Function (each input has exactly one output). Different inputs can have the same output in a function.

18. Identify the range of the set $\{(1,0),(2,0),(3,4)\}$.

    $\{0,4\}$. Range includes all unique second coordinates.

19. Identify the domain of the set $\{(-2,5),(0,1),(4,1)\}$.

    $\{-2,0,4\}$. Domain includes all unique first coordinates.

20. What is the definition of a function from a domain to a range?

    A rule that assigns each domain input exactly one range output. This defines the one-to-one correspondence property of functions.

21. Choose the word that names the output set in a function definition: domain or range?

    Range. Range refers to the output set of a function.

22. What ordered pair is plotted on the graph for an input $x$ of a function $f$?

    $(x,f(x))$. The ordered pair shows input $x$ and corresponding output $f(x)$.

23. What does the statement "$x$ is an element of the domain" guarantee about $f(x)$?

    $f(x)$ is defined and has exactly one value. Domain membership guarantees function value existence and uniqueness.

24. What word names the set of all possible outputs of a function?

    Range. Range is the set of all possible output values.

25. Find and correct the statement: "The graph of $f$ is the graph of $x=f(y)$."

    Correct: The graph of $f$ is the graph of $y=f(x)$. The corrected notation shows proper function graphing.

26. Identify the range of $f(x)=x+1$ over all real inputs.

    All real numbers. Linear functions with non-zero slope have all reals as range.

27. What test on a graph checks whether a relation is a function?

    The vertical line test. This graphical test checks the function definition property.

28. What is the meaning of the equation $y=f(x)$ in terms of a function's graph?

    It describes all points $(x,y)$ where $y$ equals the output $f(x)$. This creates a coordinate system where $x$ is input and $y$ is output.

29. What point must lie on the graph of $f$ if $f(3)=10$?

    $(3,10)$. Function values correspond to specific coordinate points.

30. Evaluate $f(2)$ for $f(x)=3x-1$.

    $5$. Substitute $x=2$: $f(2)=3(2)-1=6-1=5$.

31. What does the notation $f(x)$ mean for a function $f$ and input $x$?

    The output value of $f$ corresponding to the input $x$. Function notation represents the result of applying $f$ to input $x$.

32. Find and correct the statement: "A function can assign two outputs to one input."

    Correct: A function assigns exactly one output to each input. The corrected statement reflects the function definition.

33. Identify whether the equation $y^2=x$ represents $y$ as a function of $x$.

    Not a function (some $x$ values give two $y$ values). For example, when $x=4$, $y=±2$, giving two outputs.

34. Identify the output when $f(x)=2x+3$ and the input is $x=4$.

    $11$. Substitute $x=4$: $f(4)=2(4)+3=8+3=11$.

35. Identify whether the equation $y=x^3$ represents a function.

    Function. Each input gives exactly one output value.

36. Identify whether $\{(2,1),(3,1),(4,1),(5,1)\}$ is a function.

    Function. Each input maps to exactly one output value.

37. What is the correct reading of $f(x)$ in words?

    Function of $x$. This is the standard way to read function notation.

38. What is the domain of $f(x)=x^2-9$ when the domain is not restricted?

    All real numbers. Polynomial functions have no restrictions on inputs.

39. Identify the range of $f(x)=3$ over all real inputs.

    $\{3\}$. Constant functions have a single-element range.

40. What is the domain of $f(x)=\sqrt{x}$ over the real numbers?

    All real numbers $x$ such that $x\ge 0$. Square roots require non-negative inputs in real numbers.

41. What is the input value called when the function is written as $y=f(x)$?

    $x$ (the independent variable). The input variable determines the output value.

42. Evaluate $f(0)$ for $f(x)=5-2x$.

    $5$. Substitute $x=0$: $f(0)=5-2(0)=5-0=5$.

43. Identify the range of the relation $\{(1,2),(2,5),(3,5)\}$

    $\{2,5\}$. Range contains all second coordinates of ordered pairs.

44. Identify whether $\{(1,4),(1,7),(2,3)\}$ is a function.

    Not a function (input $1$ has two outputs). One input cannot map to multiple outputs in a function.

45. Identify whether $\{(2,1),(2,1),(3,4)\}$ is a function.

    Function (repeated pair does not create two outputs for one input). Repeated pairs don't violate the function definition.

46. Choose the word that names the input set in a function definition: domain or range?

    Domain. Domain refers to the input set of a function.

47. Identify the point on $y=f(x)$ corresponding to input $x=-2$ for $f(x)=x+4$.

    $(-2,2)$. Calculate $f(-2)=-2+4=2$, giving point $(-2,2)$.

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