Function Definitions & Notation - ACT Math

1. Identify the function that is constant for all inputs.

   $f(x) = c$, where $c$ is a constant. A constant function outputs the same value regardless of input.

2. For $f(x) = \frac{x}{x+1}$, what is the vertical asymptote?

   Vertical asymptote is $x = -1$. Vertical asymptotes occur where the denominator equals zero.

3. Simplify $f(x+1)$ if $f(x)=x^2$.

   $f(x+1)=(x+1)^2$. Replace $x$ with $(x+1)$ in the function: $f(x+1)=(x+1)^2$.

4. What is the x-intercept condition for $y=f(x)$ in function notation?

   An x-intercept occurs where $f(x)=0$. Solve for values where the function equals zero to find x-axis intersections.

5. What is $f(0)$ for $f(x) = 3x^3 - 2x$?

   $f(0) = 0$. Substitute $x = 0$: $3(0)^3 - 2(0) = 0$.

6. What does the notation $f^{-1}(x)$ represent when it exists?

   The inverse function of $f$. The inverse function reverses the input-output relationship of the original function.

7. What does the notation $f(x)-f(a)$ represent?

   The difference between outputs at inputs $x$ and $a$. This calculates the change in function values between two different input points.

8. If $f(x)=2x+1$ and $g(x)=x^2$, find $(f\circ g)(x)$.

   $(f\circ g)(x)=2x^2+1$. Substitute $g(x)=x^2$ into $f$: $f(g(x))=f(x^2)=2(x^2)+1=2x^2+1$.

9. For $f(x) = \frac{1}{x}$, what is $f(3)$?

   $f(3) = \frac{1}{3}$. Substitute $x = 3$ into the reciprocal function.

10. Define the zero of a function $f(x)$.

    Value of $x$ where $f(x) = 0$. A zero is an x-value that makes the function equal to zero.

11. If $f(x)=\frac{x+1}{2}$, what is $f^{-1}(x)$?

    $f^{-1}(x)=2x-1$. Swap $x$ and $y$, then solve for $y$: $x=\frac{y+1}{2}$ gives $y=2x-1$.

12. For $f(x) = \frac{1}{x-1}$, what is the domain?

    All real numbers except $x = 1$. Division by zero occurs when $x = 1$.

13. Define the range of the function $f(x) = x^2$.

    Non-negative real numbers. Squaring any real number produces a non-negative result.

14. Identify the range of $f(x) = \frac{1}{x}$.

    All real numbers except $0$. The reciprocal function cannot equal zero for any real input.

15. What does the notation $f(x)$ represent in a function?

    $f(x)$ is the output value of $f$ when the input is $x$. This defines function notation where $f$ is the function name and $x$ is the input variable.

16. What does the notation $f(x)+g(x)$ mean?

    A new function whose output is $f(x)+g(x)$. Function addition creates a new function by adding corresponding outputs pointwise.

17. What is the domain of a function, in words?

    The set of all allowable input values. Domain consists of all $x$-values where the function is defined and produces real outputs.

18. What is the meaning of the equation $f(x)=0$?

    The input values $x$ for which the output equals $0$. These are the zeros or roots of the function, where the graph crosses the x-axis.

19. If $f(x)=x^2$ and $g(x)=x-1$, find $(g\circ f)(2)$.

    $(g\circ f)(2)=3$. First find $f(2)=4$, then $g(4)=4-1=3$.

20. For $f(x) = x^2$, what is the vertex of the parabola?

    Vertex is $(0, 0)$. The vertex of $x^2$ occurs at the origin where the derivative is zero.

21. What does the notation $f(2) = 7$ indicate?

    The output is $7$ when input is $2$. Function notation shows the input-output relationship.

22. What is the domain restriction for $f(x)=\sqrt{x+5}$ (real numbers)?

    Domain: $x\ge -5$. The radicand $x+5$ must be non-negative for real square root values.

23. What does the notation $x\in\text{domain}(f)$ mean?

    $x$ is an allowable input for $f$. The symbol $\in$ means 'is an element of' or 'belongs to' the domain set.

24. What is the defining relationship between a function and its inverse?

    $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x$. These compositions demonstrate that inverse functions undo each other's operations.

25. What is $f(1)$ for $f(x) = 3x + 5$?

    $f(1) = 8$. Substitute $x = 1$: $3(1) + 5 = 8$.

26. Evaluate $f(0)$ if $f(x)=\frac{x-1}{x+2}$.

    $f(0)=-\frac{1}{2}$. Substitute $x=0$: $f(0)=\frac{0-1}{0+2}=\frac{-1}{2}=-\frac{1}{2}$.

27. If $f(x)=3-x$ and $g(x)=x^2$, find $(g\circ f)(x)$.

    $(g\circ f)(x)=(3-x)^2$. Substitute $f(x)=3-x$ into $g$: $g(f(x))=g(3-x)=(3-x)^2$.

28. What does the notation $y\in\text{range}(f)$ mean?

    $y$ is a possible output value of $f$. The symbol $\in$ means the output value exists within the function's range set.

29. For $f(x) = 3x + 2$, what is $f(4)$?

    $f(4) = 14$. Substitute $x = 4$ into the function: $3(4) + 2 = 14$.

30. What does it mean if $f(a)\neq f(b)$?

    Inputs $a$ and $b$ produce different outputs. This indicates the function produces distinct outputs for these different inputs.

31. What is the slope of $f(x) = 7x - 4$?

    Slope is $7$. Linear functions have the form $mx + b$ where $m$ is the slope.

32. Evaluate $f(3)$ if $f(x)=2x-5$.

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

33. Simplify $f(x)-f(1)$ if $f(x)=x^2+2x$.

    $f(x)-f(1)=x^2+2x-3$. Calculate $f(x)=(x^2+2x)$ and $f(1)=3$, so the difference is $x^2+2x-3$.

34. Find the y-intercept of $f(x) = x^2 + 3x + 2$.

    The point $(0, 2)$. Set $x = 0$ to find where the graph crosses the y-axis.

35. Find the inverse of $f(x) = 2x + 3$.

    $f^{-1}(x) = \frac{x-3}{2}$. Solve $y = 2x + 3$ for $x$, then swap variables.

36. What is the range of $f(x) = \text{abs}(x)$?

    Non-negative real numbers. Absolute value function always produces non-negative outputs.

37. Find $f(2)$ if $f(x) = 2x^2 - 3x + 1$.

    $f(2) = 3$. Substitute $x = 2$: $2(4) - 3(2) + 1 = 8 - 6 + 1 = 3$.

38. What is the notation for the composition of $f$ and $g$?

    $(f \, \text{∘} \, g)(x)$. This symbol represents function composition, where one function is applied to another.

39. What does the notation $f(x)=\sqrt{x}$ imply about the domain (real numbers)?

    Domain: $x\ge 0$. Square root requires non-negative inputs to produce real number outputs.

40. If $f(x)=x-5$, what is $f^{-1}(x)$?

    $f^{-1}(x)=x+5$. For linear functions, the inverse reverses the operation: subtract becomes add.

41. What is $f(3)$ if $f(x) = x^2 - 4x + 4$?

    $f(3) = 1$. Substitute $x = 3$: $(3)^2 - 4(3) + 4 = 9 - 12 + 4 = 1$.

42. Simplify $f(x) = \frac{x^2 - 1}{x - 1}$.

    $f(x) = x + 1$, $x \neq 1$. Factor and cancel, but note the domain restriction.

43. Find $f(x+1)$ for $f(x) = x^2 - x$.

    $f(x+1) = x^2 + x$. Replace $x$ with $x+1$ in the function: $(x+1)^2 - (x+1)$.

44. What is the notation for the absolute value of $x$?

    $|x|$. Vertical bars around a variable indicate absolute value.

45. What does it mean if $f(a)=f(b)$?

    Inputs $a$ and $b$ produce the same output. Different inputs can map to the same output value in many-to-one relationships.

46. What is $f(-3)$ for $f(x) = 2x + 1$?

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

47. What is the domain of $f(x) = \frac{1}{x^2 - 4}$?

    All real numbers except $x = \text{±}2$. Denominator equals zero when $x^2 = 4$, so $x = ±2$.

48. Find $x$ if $f(x)=x^2-9$ and $f(x)=0$.

    $x=3$ or $x=-3$. Set $x^2-9=0$, so $x^2=9$, giving $x=\pm 3$.

49. What is the domain restriction for $f(x)=\frac{1}{x-3}$ (real numbers)?

    Domain: $x\ne 3$. Set the denominator $x-3\neq 0$ to avoid division by zero.

50. What is the correct interpretation of $f^{-1}(x)$ (not a power)?

    $f^{-1}(x)$ is the inverse function, not $\frac{1}{f(x)}$. The superscript $-1$ denotes inverse function, not reciprocal or negative exponent.

51. Identify the notation for the reciprocal of a function output.

    $\frac{1}{f(x)}$. This represents one divided by the function output, the multiplicative inverse.

52. Evaluate $f(-2)$ if $f(x)=x^2+4$.

    $f(-2)=8$. Substitute $x=-2$: $f(-2)=(-2)^2+4=4+4=8$.

53. If $f(x) = x^2$ and $g(x) = 2x$, find $(f \, \text{∘} \, g)(x)$.

    $(f \, \text{∘} \, g)(x) = 4x^2$. First apply $g$, then $f$: $f(g(x)) = f(2x) = (2x)^2 = 4x^2$.

54. What is the y-intercept of a function $y=f(x)$ in function notation?

    The y-intercept is $f(0)$. Set $x=0$ and evaluate the function to find where the graph crosses the y-axis.

55. What is the standard meaning of the statement $y=f(x)$?

    $y$ equals the output of function $f$ for input $x$. This is the standard function equation format relating dependent and independent variables.

56. What does $f(a)$ represent if $f(x)$ is a function?

    Value of $f(x)$ at $x = a$. Function notation indicates the output when input is $a$.

57. What is the domain of the function $f(x) = \frac{1}{x-2}$?

    All real numbers except $x = 2$. Division by zero occurs when $x = 2$, making the function undefined.

58. What is $f(0)$ for $f(x) = 5x - 7$?

    $f(0) = -7$. Substitute $x = 0$: $5(0) - 7 = -7$.

59. What does the notation $f(a)$ mean?

    The value of the function when $x=a$. Substitute $a$ for $x$ in the function definition to get the specific output.

60. What does the notation $f(x+h)$ indicate compared with $f(x)$?

    The output when the input is shifted to $x+h$. The input is horizontally shifted by $h$ units, affecting the function's evaluation point.

61. Express $f(x) = x^2 + 2x + 1$ in vertex form.

    $f(x) = (x+1)^2$. Factor the perfect square trinomial to vertex form.

62. What is the vertical intercept of $f(x) = 4x + 6$?

    The point $(0, 6)$. Set $x = 0$ to find the y-intercept: $f(0) = 6$.

63. If $f(x)=x+4$ and $g(x)=2x$, find $(f\circ g)(3)$.

    $(f\circ g)(3)=10$. First find $g(3)=6$, then $f(6)=6+4=10$.

64. What does the notation $(f\circ g)(x)$ mean?

    $(f\circ g)(x)=f(g(x))$. Composition means applying $g$ first, then applying $f$ to that result.

65. Find $x$ if $f(x)=2x-7$ and $f(x)=5$.

    $x=6$. Set $2x-7=5$ and solve: $2x=12$, so $x=6$.

66. What is the value of $f(-5)$ if $f(x) = -x + 6$?

    $f(-5) = 11$. Substitute $x = -5$: $-(-5) + 6 = 5 + 6 = 11$.

67. Find $f(a)$ if $f(x)=3x^2-2x$.

    $f(a)=3a^2-2a$. Replace every occurrence of $x$ with $a$ in the function expression.

68. What does the notation $|f(x)|$ mean?

    The absolute value of the output $f(x)$. Takes the absolute value of whatever the function outputs for input $x$.

69. Simplify $f(2x)$ if $f(x)=x^2-3$.

    $f(2x)=4x^2-3$. Replace $x$ with $2x$: $f(2x)=(2x)^2-3=4x^2-3$.

70. What does $f^{-1}(x)$ represent in function notation?

    Inverse of function $f(x)$. The inverse function undoes the original function's operation.

71. For $f(x) = x^3 - x$, find $f(-2)$.

    $f(-2) = -6$. Substitute $x = -2$: $(-2)^3 - (-2) = -8 + 2 = -6$.

72. What does the notation $f(x)=\frac{1}{x}$ imply about the domain (real numbers)?

    Domain: $x\ne 0$. Division by zero is undefined, so the denominator cannot equal zero.

73. What is the difference between $f(x)^2$ and $f(x^2)$?

    $f(x)^2$ squares the output; $f(x^2)$ inputs $x^2$. The first squares the result; the second evaluates the function at $x^2$.

74. What is the range of a function, in words?

    The set of all possible output values. Range consists of all $y$-values that the function can actually produce as outputs.

75. What is a function, stated using the input-output rule?

    Each input has exactly one output. This is the fundamental definition ensuring well-defined mathematical relationships.

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