ACT Math Flashcards: Function Definitions And Notation

What this deck covers

This deck focuses on Function Definitions And Notation, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

Flashcard 5: What does the notation $x\in\text{domain}(f)$ mean?

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

Flashcard 6: What is $f(0)$ for $f(x) = 3x^3 - 2x$?

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

Flashcard 7: What does the notation $f^{-1}(x)$ represent when it exists?

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

Flashcard 8: What is $f(0)$ for $f(x) = 5x - 7$?

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

Flashcard 9: For $f(x) = x^3 - x$, find $f(-2)$.

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

Flashcard 10: What does the notation $f(x)-f(a)$ represent?

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

Flashcard 11: If $f(x)=2x+1$ and $g(x)=x^2$, find $(f\circ g)(x)$.

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

Flashcard 12: For $f(x) = \frac{1}{x}$, what is $f(3)$?

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

Flashcard 13: Define the zero of a function $f(x)$.

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

Flashcard 14: Define the range of the function $f(x) = x^2$.

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

Flashcard 15: If $f(x)=3-x$ and $g(x)=x^2$, find $(g\circ f)(x)$.

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

Flashcard 16: If $f(x)=\frac{x+1}{2}$, what is $f^{-1}(x)$?

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

Flashcard 17: Find $f(a)$ if $f(x)=3x^2-2x$.

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

Flashcard 18: For $f(x) = \frac{1}{x-1}$, what is the domain?

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

Flashcard 19: Define the range of the function $f(x) = x^2$.

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

Flashcard 20: Identify the range of $f(x) = \frac{1}{x}$.

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

Flashcard 21: What does the notation $f(x)$ represent in a function?

Answer: $f(x)$ 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.

Flashcard 22: What does the notation $f(x)+g(x)$ mean?

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

Flashcard 23: Evaluate $f(-2)$ if $f(x)=x^2+4$.

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

Flashcard 24: What is the range of a function, in words?

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

Flashcard 25: What is the correct interpretation of $f^{-1}(x)$ (not a power)?

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

Flashcard 26: What is the domain of a function, in words?

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

Flashcard 27: What is the domain of $f(x) = \frac{1}{x^2 - 4}$?

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

Flashcard 28: What is $f(-3)$ for $f(x) = 2x + 1$?

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

Flashcard 29: What does the notation $f(x)-f(a)$ represent?

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

Flashcard 30: What is the defining relationship between a function and its inverse?

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

Flashcard 31: What is the meaning of the equation $f(x)=0$?

Answer: 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.

Flashcard 32: What does the notation $f(2) = 7$ indicate?

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

Flashcard 33: What is the domain of a function, in words?

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

Flashcard 34: What is the meaning of the equation $f(x)=0$?

Answer: 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.

Flashcard 35: If $f(x)=x^2$ and $g(x)=x-1$, find $(g\circ f)(2)$.

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

Flashcard 36: For $f(x) = x^2$, what is the vertex of the parabola?

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

Flashcard 37: What is the y-intercept of a function $y=f(x)$ in function notation?

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

Flashcard 38: What is the standard meaning of the statement $y=f(x)$?

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

Flashcard 39: What does the notation $y\in\text{range}(f)$ mean?

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

Flashcard 40: If $f(x) = x^2$ and $g(x) = 2x$, find $(f \, \text{∘} \, g)(x)$.

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

Flashcard 41: What does the notation $f(2) = 7$ indicate?

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

Flashcard 42: What is the domain restriction for $f(x)=\sqrt{x+5}$ (real numbers)?

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

Flashcard 43: What does the notation $x\in\text{domain}(f)$ mean?

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

Flashcard 44: What does the notation $f(x+h)$ indicate compared with $f(x)$?

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

Flashcard 45: What is the defining relationship between a function and its inverse?

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

Flashcard 46: Find $x$ if $f(x)=x^2-9$ and $f(x)=0$.

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

Flashcard 47: What does it mean if $f(a)\neq f(b)$?

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

Flashcard 48: What is $f(1)$ for $f(x) = 3x + 5$?

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

Flashcard 49: If $f(x)=\frac{x+1}{2}$, what is $f^{-1}(x)$?

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

Flashcard 50: Evaluate $f(0)$ if $f(x)=\frac{x-1}{x+2}$.

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

Flashcard 51: If $f(x)=3-x$ and $g(x)=x^2$, find $(g\circ f)(x)$.

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

Flashcard 52: What is $f(3)$ if $f(x) = x^2 - 4x + 4$?

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

Flashcard 53: What does the notation $y\in\text{range}(f)$ mean?

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

Flashcard 54: For $f(x) = 3x + 2$, what is $f(4)$?

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

Flashcard 55: What does it mean if $f(a)\neq f(b)$?

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

Flashcard 56: Evaluate $f(0)$ if $f(x)=\frac{x-1}{x+2}$.

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

Flashcard 57: Simplify $f(x)-f(1)$ if $f(x)=x^2+2x$.

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

Flashcard 58: What does $f(a)$ represent if $f(x)$ is a function?

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

Flashcard 59: What is the slope of $f(x) = 7x - 4$?

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

Flashcard 60: Evaluate $f(3)$ if $f(x)=2x-5$.

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

Flashcard 61: Identify the notation for the reciprocal of a function output.

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

Flashcard 62: Simplify $f(x)-f(1)$ if $f(x)=x^2+2x$.

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

Flashcard 63: What does the notation $|f(x)|$ mean?

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

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

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

Flashcard 65: Find the y-intercept of $f(x) = x^2 + 3x + 2$.

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

Flashcard 66: For $f(x) = \frac{1}{x-1}$, what is the domain?

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

Flashcard 67: What is $f(0)$ for $f(x) = 3x^3 - 2x$?

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

Flashcard 68: Find the inverse of $f(x) = 2x + 3$.

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

Flashcard 69: What is the range of $f(x) = \text{abs}(x)$?

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

Flashcard 70: Find $f(2)$ if $f(x) = 2x^2 - 3x + 1$.

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

Flashcard 71: What is the value of $f(-5)$ if $f(x) = -x + 6$?

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

Flashcard 72: What is the notation for the composition of $f$ and $g$?

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

Flashcard 73: What is the notation for the composition of $f$ and $g$?

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

Flashcard 74: What does the notation $f(x)=\sqrt{x}$ imply about the domain (real numbers)?

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

Flashcard 75: Express $f(x) = x^2 + 2x + 1$ in vertex form.

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

Flashcard 76: What does the notation $f(x)$ represent in a function?

Answer: $f(x)$ 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.

Flashcard 77: Identify the range of $f(x) = \frac{1}{x}$.

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

Flashcard 78: If $f(x)=x-5$, what is $f^{-1}(x)$?

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

Flashcard 79: What is $f(3)$ if $f(x) = x^2 - 4x + 4$?

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

Flashcard 80: For $f(x) = \frac{x}{x+1}$, what is the vertical asymptote?

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

Flashcard 81: Simplify $f(2x)$ if $f(x)=x^2-3$.

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

Flashcard 82: Find $f(2)$ if $f(x) = 2x^2 - 3x + 1$.

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

Flashcard 83: Simplify $f(x) = \frac{x^2 - 1}{x - 1}$.

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

Flashcard 84: What is the slope of $f(x) = 7x - 4$?

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

Flashcard 85: If $f(x)=x+4$ and $g(x)=2x$, find $(f\circ g)(3)$.

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

Flashcard 86: Find $f(x+1)$ for $f(x) = x^2 - x$.

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

Flashcard 87: What is the notation for the absolute value of $x$?

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

Flashcard 88: What does the notation $f^{-1}(x)$ represent when it exists?

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

Flashcard 89: What does it mean if $f(a)=f(b)$?

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

Flashcard 90: What is $f(-3)$ for $f(x) = 2x + 1$?

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

Flashcard 91: What is the domain restriction for $f(x)=\sqrt{x+5}$ (real numbers)?

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

Flashcard 92: What is the domain of $f(x) = \frac{1}{x^2 - 4}$?

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

Flashcard 93: Find $x$ if $f(x)=x^2-9$ and $f(x)=0$.

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

Flashcard 94: What is the domain restriction for $f(x)=\frac{1}{x-3}$ (real numbers)?

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

Flashcard 95: What is the domain restriction for $f(x)=\frac{1}{x-3}$ (real numbers)?

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

Flashcard 96: What is the correct interpretation of $f^{-1}(x)$ (not a power)?

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

Flashcard 97: Identify the notation for the reciprocal of a function output.

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

Flashcard 98: Simplify $f(x+1)$ if $f(x)=x^2$.

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

Flashcard 99: Find $f(x+1)$ for $f(x) = x^2 - x$.

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

Flashcard 100: Evaluate $f(-2)$ if $f(x)=x^2+4$.

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