Function Definitions & Notation - ACT Math
Card 1 of 30
What is $f(0)$ for $f(x) = 5x - 7$?
What is $f(0)$ for $f(x) = 5x - 7$?
Tap to reveal answer
$f(0) = -7$. Substitute $x = 0$: $5(0) - 7 = -7$.
$f(0) = -7$. Substitute $x = 0$: $5(0) - 7 = -7$.
← Didn't Know|Knew It →
What does it mean if $f(a)=f(b)$?
What does it mean if $f(a)=f(b)$?
Tap to reveal answer
Inputs $a$ and $b$ produce the same output. Different inputs can map to the same output value in many-to-one relationships.
Inputs $a$ and $b$ produce the same output. Different inputs can map to the same output value in many-to-one relationships.
← Didn't Know|Knew It →
What is the domain of $f(x) = \frac{1}{x^2 - 4}$?
What is the domain of $f(x) = \frac{1}{x^2 - 4}$?
Tap to reveal answer
All real numbers except $x = \text{±}2$. Denominator equals zero when $x^2 = 4$, so $x = ±2$.
All real numbers except $x = \text{±}2$. Denominator equals zero when $x^2 = 4$, so $x = ±2$.
← Didn't Know|Knew It →
For $f(x) = 3x + 2$, what is $f(4)$?
For $f(x) = 3x + 2$, what is $f(4)$?
Tap to reveal answer
$f(4) = 14$. Substitute $x = 4$ into the function: $3(4) + 2 = 14$.
$f(4) = 14$. Substitute $x = 4$ into the function: $3(4) + 2 = 14$.
← Didn't Know|Knew It →
What does the notation $|f(x)|$ mean?
What does the notation $|f(x)|$ mean?
Tap to reveal answer
The absolute value of the output $f(x)$. Takes the absolute value of whatever the function outputs for input $x$.
The absolute value of the output $f(x)$. Takes the absolute value of whatever the function outputs for input $x$.
← Didn't Know|Knew It →
What does the notation $x\in\text{domain}(f)$ mean?
What does the notation $x\in\text{domain}(f)$ mean?
Tap to reveal answer
$x$ is an allowable input for $f$. The symbol $\in$ means 'is an element of' or 'belongs to' the domain set.
$x$ is an allowable input for $f$. The symbol $\in$ means 'is an element of' or 'belongs to' the domain set.
← Didn't Know|Knew It →
What does the notation $f(x)=\frac{1}{x}$ imply about the domain (real numbers)?
What does the notation $f(x)=\frac{1}{x}$ imply about the domain (real numbers)?
Tap to reveal answer
Domain: $x\ne 0$. Division by zero is undefined, so the denominator cannot equal zero.
Domain: $x\ne 0$. Division by zero is undefined, so the denominator cannot equal zero.
← Didn't Know|Knew It →
What is the y-intercept of a function $y=f(x)$ in function notation?
What is the y-intercept of a function $y=f(x)$ in function notation?
Tap to reveal answer
The y-intercept is $f(0)$. Set $x=0$ and evaluate the function to find where the graph crosses the y-axis.
The y-intercept is $f(0)$. Set $x=0$ and evaluate the function to find where the graph crosses the y-axis.
← Didn't Know|Knew It →
What does the notation $f^{-1}(x)$ represent when it exists?
What does the notation $f^{-1}(x)$ represent when it exists?
Tap to reveal answer
The inverse function of $f$. The inverse function reverses the input-output relationship of the original function.
The inverse function of $f$. The inverse function reverses the input-output relationship of the original function.
← Didn't Know|Knew It →
Identify the notation for the reciprocal of a function output.
Identify the notation for the reciprocal of a function output.
Tap to reveal answer
$\frac{1}{f(x)}$. This represents one divided by the function output, the multiplicative inverse.
$\frac{1}{f(x)}$. This represents one divided by the function output, the multiplicative inverse.
← Didn't Know|Knew It →
Evaluate $f(-2)$ if $f(x)=x^2+4$.
Evaluate $f(-2)$ if $f(x)=x^2+4$.
Tap to reveal answer
$f(-2)=8$. Substitute $x=-2$: $f(-2)=(-2)^2+4=4+4=8$.
$f(-2)=8$. Substitute $x=-2$: $f(-2)=(-2)^2+4=4+4=8$.
← Didn't Know|Knew It →
Evaluate $f(0)$ if $f(x)=\frac{x-1}{x+2}$.
Evaluate $f(0)$ if $f(x)=\frac{x-1}{x+2}$.
Tap to reveal answer
$f(0)=-\frac{1}{2}$. Substitute $x=0$: $f(0)=\frac{0-1}{0+2}=\frac{-1}{2}=-\frac{1}{2}$.
$f(0)=-\frac{1}{2}$. Substitute $x=0$: $f(0)=\frac{0-1}{0+2}=\frac{-1}{2}=-\frac{1}{2}$.
← Didn't Know|Knew It →
Find $f(a)$ if $f(x)=3x^2-2x$.
Find $f(a)$ if $f(x)=3x^2-2x$.
Tap to reveal answer
$f(a)=3a^2-2a$. Replace every occurrence of $x$ with $a$ in the function expression.
$f(a)=3a^2-2a$. Replace every occurrence of $x$ with $a$ in the function expression.
← Didn't Know|Knew It →
Find $x$ if $f(x)=x^2-9$ and $f(x)=0$.
Find $x$ if $f(x)=x^2-9$ and $f(x)=0$.
Tap to reveal answer
$x=3$ or $x=-3$. Set $x^2-9=0$, so $x^2=9$, giving $x=\pm 3$.
$x=3$ or $x=-3$. Set $x^2-9=0$, so $x^2=9$, giving $x=\pm 3$.
← Didn't Know|Knew It →
Find the y-intercept of $f(x) = x^2 + 3x + 2$.
Find the y-intercept of $f(x) = x^2 + 3x + 2$.
Tap to reveal answer
The point $(0, 2)$. Set $x = 0$ to find where the graph crosses the y-axis.
The point $(0, 2)$. Set $x = 0$ to find where the graph crosses the y-axis.
← Didn't Know|Knew It →
For $f(x) = \frac{x}{x+1}$, what is the vertical asymptote?
For $f(x) = \frac{x}{x+1}$, what is the vertical asymptote?
Tap to reveal answer
Vertical asymptote is $x = -1$. Vertical asymptotes occur where the denominator equals zero.
Vertical asymptote is $x = -1$. Vertical asymptotes occur where the denominator equals zero.
← Didn't Know|Knew It →
What is the range of $f(x) = \text{abs}(x)$?
What is the range of $f(x) = \text{abs}(x)$?
Tap to reveal answer
Non-negative real numbers. Absolute value function always produces non-negative outputs.
Non-negative real numbers. Absolute value function always produces non-negative outputs.
← Didn't Know|Knew It →
Identify the range of $f(x) = \frac{1}{x}$.
Identify the range of $f(x) = \frac{1}{x}$.
Tap to reveal answer
All real numbers except $0$. The reciprocal function cannot equal zero for any real input.
All real numbers except $0$. The reciprocal function cannot equal zero for any real input.
← Didn't Know|Knew It →
What is the domain of $f(x) = \frac{1}{x^2 - 4}$?
What is the domain of $f(x) = \frac{1}{x^2 - 4}$?
Tap to reveal answer
All real numbers except $x = \text{±}2$. Denominator equals zero when $x^2 = 4$, so $x = ±2$.
All real numbers except $x = \text{±}2$. Denominator equals zero when $x^2 = 4$, so $x = ±2$.
← Didn't Know|Knew It →
Simplify $f(x) = \frac{x^2 - 1}{x - 1}$.
Simplify $f(x) = \frac{x^2 - 1}{x - 1}$.
Tap to reveal answer
$f(x) = x + 1$, $x \neq 1$. Factor and cancel, but note the domain restriction.
$f(x) = x + 1$, $x \neq 1$. Factor and cancel, but note the domain restriction.
← Didn't Know|Knew It →
For $f(x) = \frac{1}{x-1}$, what is the domain?
For $f(x) = \frac{1}{x-1}$, what is the domain?
Tap to reveal answer
All real numbers except $x = 1$. Division by zero occurs when $x = 1$.
All real numbers except $x = 1$. Division by zero occurs when $x = 1$.
← Didn't Know|Knew It →
What does the notation $f(2) = 7$ indicate?
What does the notation $f(2) = 7$ indicate?
Tap to reveal answer
The output is $7$ when input is $2$. Function notation shows the input-output relationship.
The output is $7$ when input is $2$. Function notation shows the input-output relationship.
← Didn't Know|Knew It →
Find the inverse of $f(x) = 2x + 3$.
Find the inverse of $f(x) = 2x + 3$.
Tap to reveal answer
$f^{-1}(x) = \frac{x-3}{2}$. Solve $y = 2x + 3$ for $x$, then swap variables.
$f^{-1}(x) = \frac{x-3}{2}$. Solve $y = 2x + 3$ for $x$, then swap variables.
← Didn't Know|Knew It →
What is the value of $f(-5)$ if $f(x) = -x + 6$?
What is the value of $f(-5)$ if $f(x) = -x + 6$?
Tap to reveal answer
$f(-5) = 11$. Substitute $x = -5$: $-(-5) + 6 = 5 + 6 = 11$.
$f(-5) = 11$. Substitute $x = -5$: $-(-5) + 6 = 5 + 6 = 11$.
← Didn't Know|Knew It →
What is $f(3)$ if $f(x) = x^2 - 4x + 4$?
What is $f(3)$ if $f(x) = x^2 - 4x + 4$?
Tap to reveal answer
$f(3) = 1$. Substitute $x = 3$: $(3)^2 - 4(3) + 4 = 9 - 12 + 4 = 1$.
$f(3) = 1$. Substitute $x = 3$: $(3)^2 - 4(3) + 4 = 9 - 12 + 4 = 1$.
← Didn't Know|Knew It →
For $f(x) = x^2$, what is the vertex of the parabola?
For $f(x) = x^2$, what is the vertex of the parabola?
Tap to reveal answer
Vertex is $(0, 0)$. The vertex of $x^2$ occurs at the origin where the derivative is zero.
Vertex is $(0, 0)$. The vertex of $x^2$ occurs at the origin where the derivative is zero.
← Didn't Know|Knew It →
What is $f(-3)$ for $f(x) = 2x + 1$?
What is $f(-3)$ for $f(x) = 2x + 1$?
Tap to reveal answer
$f(-3) = -5$. Substitute $x = -3$: $2(-3) + 1 = -6 + 1 = -5$.
$f(-3) = -5$. Substitute $x = -3$: $2(-3) + 1 = -6 + 1 = -5$.
← Didn't Know|Knew It →
What is the domain of the function $f(x) = \frac{1}{x-2}$?
What is the domain of the function $f(x) = \frac{1}{x-2}$?
Tap to reveal answer
All real numbers except $x = 2$. Division by zero occurs when $x = 2$, making the function undefined.
All real numbers except $x = 2$. Division by zero occurs when $x = 2$, making the function undefined.
← Didn't Know|Knew It →
Define the range of the function $f(x) = x^2$.
Define the range of the function $f(x) = x^2$.
Tap to reveal answer
Non-negative real numbers. Squaring any real number produces a non-negative result.
Non-negative real numbers. Squaring any real number produces a non-negative result.
← Didn't Know|Knew It →
For $f(x) = 3x + 2$, what is $f(4)$?
For $f(x) = 3x + 2$, what is $f(4)$?
Tap to reveal answer
$f(4) = 14$. Substitute $x = 4$ into the function: $3(4) + 2 = 14$.
$f(4) = 14$. Substitute $x = 4$ into the function: $3(4) + 2 = 14$.
← Didn't Know|Knew It →