Algebra Flashcards: Function Notation And Evaluation

What this deck covers

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

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: What does it mean if an input $x$ is not in the domain of $f$?

Answer: $f(x)$ is undefined for that input. The function cannot be evaluated at that input.

Flashcard 2: Identify whether $f(2)$ is defined if $f(x)=\frac{1}{x-2}$.

Answer: $f(2)$ is undefined. Division by zero occurs when $x=2$.

Flashcard 3: What is the domain of $f(x)=\frac{1}{x+3}$ in set notation?

Answer: All real $x$ such that $x\neq -3$. Denominator cannot equal zero, so exclude $x=-3$.

Flashcard 4: What is the domain of a function in words?

Answer: The set of all allowable input values. Only these values can be substituted into the function.

Flashcard 5: What is the domain of $f(x)=\sqrt{x-5}$ in inequality form?

Answer: $x\ge 5$. Square root requires non-negative radicand.

Flashcard 6: Find the domain of $f(x)=\frac{2}{x}$ in restriction form.

Answer: All real $x$ such that $x\neq 0$. Denominator cannot equal zero.

Flashcard 7: Interpret $P(5)=1200$ if $P(t)$ is population after $t$ years.

Answer: After $5$ years, the population is $1200$. Function output shows population at given time.

Flashcard 8: Find $p(x-1)$ if $p(x)=x^2$.

Answer: $p(x-1)=(x-1)^2$. Replace $x$ with $(x-1)$: $p(x-1)=(x-1)^2$.

Flashcard 9: Find $f(-2)$ if $f(x)=-x+6$.

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

Flashcard 10: What does the statement $f(x)=0$ ask you to find?

Answer: All inputs $x$ that make the output equal to $0$. Find the zeros or roots of the function.

Flashcard 11: What is the difference between $f(x)$ and $f\,x$ in algebra?

Answer: $f(x)$ is a function value; $f\,x$ is multiplication. Parentheses indicate function evaluation, not multiplication.

Flashcard 12: Find $q(-2)$ if $q(x)=x^2+4x+1$.

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

Flashcard 13: Find $s(3)$ if $s(x)=\frac{x+1}{2}$.

Answer: $s(3)=2$. Substitute $x=3$: $s(3)=\frac{3+1}{2}=2$.

Flashcard 14: Find $g(x+2)$ if $g(x)=3x-1$.

Answer: $g(x+2)=3x+5$. Replace $x$ with $(x+2)$: $g(x+2)=3(x+2)-1=3x+5$.

Flashcard 15: What does the statement $f(x)<0$ describe?

Answer: Inputs $x$ where the function output is negative. The function is below the $x$-axis at these inputs.

Flashcard 16: Find $f(-3)$ if $f(x)=x^2-1$.

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

Flashcard 17: Find $p(1)$ if $p(x)=3x^2+2x$.

Answer: $p(1)=5$. Substitute $x=1$: $p(1)=3(1)^2+2(1)=5$.

Flashcard 18: Find $r(5)$ if $r(x)=10-2x$.

Answer: $r(5)=0$. Substitute $x=5$: $r(5)=10-2(5)=0$.

Flashcard 19: What does the statement $f(x)>0$ describe?

Answer: Inputs $x$ where the function output is positive. The function is above the $x$-axis at these inputs.

Flashcard 20: Find $h(4)$ if $h(x)=2(x-1)$.

Answer: $h(4)=6$. Substitute $x=4$: $h(4)=2(4-1)=6$.

Flashcard 21: Interpret $C(10)=25$ if $C(d)$ is the cost in dollars for $d$ miles.

Answer: A $10$-mile trip costs $25$ dollars. Function notation describes real-world relationships.

Flashcard 22: What does $f(x)-f(a)$ represent?

Answer: The difference between outputs at inputs $x$ and $a$. Shows the change in function values between inputs.

Flashcard 23: What is the difference between $f(x)$ and $f\cdot x$?

Answer: $f(x)$ is a function value; $f\cdot x$ is multiplication. Parentheses indicate function evaluation, not multiplication.

Flashcard 24: Find $g(x+2)$ if $g(x)=3x-1$.

Answer: $g(x+2)=3x+5$. Replace $x$ with $(x+2)$: $g(x+2)=3(x+2)-1=3x+5$.

Flashcard 25: What is the range of a function in words?

Answer: The set of all possible output values. All values the function can produce as outputs.

Flashcard 26: Find $t(-1)$ if $t(x)=\lvert x\rvert+2$.

Answer: $t(-1)=3$. Substitute $x=-1$: $t(-1)=\lvert-1\rvert+2=3$.

Flashcard 27: Find $f(2)$ if the table gives $f(2)=9$.

Answer: $f(2)=9$. Read function value directly from table.

Flashcard 28: Find the domain of $f(x)=\sqrt{x+4}$ in inequality form.

Answer: $x\ge -4$. Square root requires $x+4\ge 0$.

Flashcard 29: Find $h(2x)$ if $h(x)=x^2-4$.

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

Flashcard 30: Find $q(1)$ if $q(x)=\frac{4}{x+1}$.

Answer: $q(1)=2$. Substitute $x=1$: $q(1)=\frac{4}{1+1}=2$.

Flashcard 31: Find $g(2)$ if $g(x)=\frac{1}{2}x-4$.

Answer: $g(2)=-3$. Substitute $x=2$: $g(2)=\frac{1}{2}(2)-4=-3$.

Flashcard 32: What is the domain of $f(x)=\frac{1}{x+3}$ in set notation?

Answer: All real $x$ such that $x\neq -3$. Denominator cannot equal zero, so exclude $x=-3$.

Flashcard 33: What is the meaning of the statement $f(3)=7$?

Answer: When $x=3$, the function output is $7$. Substitute $x=3$ into the function to get output $7$.

Flashcard 34: What does $f(0)$ represent on the graph of $y=f(x)$?

Answer: The $y$-intercept value. When $x=0$, output gives $y$-intercept.

Flashcard 35: Identify whether $f(-1)$ is defined if $f(x)=\sqrt{x+1}$.

Answer: $f(-1)$ is defined and equals $0$. Square root of $0$ is defined: $f(-1)=\sqrt{0}=0$.

Flashcard 36: Identify the input and output in the ordered pair $(x,f(x))$.

Answer: Input is $x$; output is $f(x)$. First coordinate is input, second is output.

Flashcard 37: Find $u(-4)$ if $u(x)=\lvert x+1\rvert$.

Answer: $u(-4)=3$. Substitute $x=-4$: $u(-4)=\lvert-4+1\rvert=3$.

Flashcard 38: What is the domain of $f(x)=\sqrt{2-x}$ in inequality form?

Answer: $x\le 2$. Square root requires non-negative radicand.

Flashcard 39: Find $h(0)$ if $h(x)=5-3x$.

Answer: $h(0)=5$. Substitute $x=0$: $h(0)=5-3(0)=5$.

Flashcard 40: What is the domain of $f(x)=\frac{5}{x^2-9}$ in restriction form?

Answer: All real $x$ such that $x\neq 3$ and $x\neq -3$. Factor: $x^2-9=(x-3)(x+3)$, exclude both zeros.

Flashcard 41: What point corresponds to $f(a)=b$ on the graph of $y=f(x)$?

Answer: The point $(a,b)$. Function notation corresponds to coordinate pairs.

Flashcard 42: What is the meaning of the statement $f(a)=b$?

Answer: When $x=a$, the function output is $b$. General form showing input $a$ produces output $b$.

Flashcard 43: Find $f(4)$ if $f(x)=2x+5$.

Answer: $f(4)=13$. Substitute $x=4$: $f(4)=2(4)+5=13$.

Flashcard 44: Find the domain of $f(x)=\sqrt{x}$ in inequality form.

Answer: $x\ge 0$. Square root requires non-negative radicand.

Flashcard 45: Find $p(0)$ if $p(x)=x^3-2x$.

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

Flashcard 46: Find $p(x-1)$ if $p(x)=x^2$.

Answer: $p(x-1)=(x-1)^2$. Replace $x$ with $(x-1)$: $p(x-1)=(x-1)^2$.

Flashcard 47: Find $g(6)$ if $g(x)=\frac{x}{3}+1$.

Answer: $g(6)=3$. Substitute $x=6$: $g(6)=\frac{6}{3}+1=3$.

Flashcard 48: Find the domain of $f(x)=\frac{x+1}{x-7}$ in restriction form.

Answer: All real $x$ such that $x\neq 7$. Denominator cannot equal zero.

Flashcard 49: What does $f(x+h)$ mean in function notation?

Answer: The output when the input is $x+h$. Substitute the entire expression $(x+h)$ for $x$.

Flashcard 50: Interpret $T(3)=68$ if $T(h)$ is temperature after $h$ hours.

Answer: After $3$ hours, the temperature is $68$ degrees. Function value represents temperature at specific time.

Flashcard 51: Find $p(0)$ if $p(x)=x^3-2x$.

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

Flashcard 52: Find $t(-1)$ if $t(x)=\lvert x\rvert+2$.

Answer: $t(-1)=3$. Substitute $x=-1$: $t(-1)=\lvert-1\rvert+2=3$.

Flashcard 53: Find the domain of $f(x)=\frac{1}{(x-1)(x+2)}$ in restriction form.

Answer: All real $x$ such that $x\neq 1$ and $x\neq -2$. Both factors in denominator cannot equal zero.

Flashcard 54: Find $f(a)$ if $f(x)=x^2+3$.

Answer: $f(a)=a^2+3$. Replace $x$ with $a$ in the function expression.

Flashcard 55: What is the standard notation for the output variable of $f(x)$?

Answer: $y=f(x)$. Standard way to represent function output.

Flashcard 56: Interpret $d(2.5)=150$ if $d(t)$ is distance in miles after $t$ hours.

Answer: After $2.5$ hours, the distance traveled is $150$ miles. Function describes distance as function of time.

Flashcard 57: What does the notation $f(x)$ represent in function notation?

Answer: $f(x)$ is the output value of function $f$ for input $x$. Function notation shows the input-output relationship.

Flashcard 58: Find $f(a)$ if $f(x)=x^2+3$.

Answer: $f(a)=a^2+3$. Replace $x$ with $a$ in the function expression.

Flashcard 59: Identify $x$ if $f(x)=12$ and the table shows $f(4)=12$.

Answer: $x=4$. Find input that produces given output.

Flashcard 60: Find $m(3)$ if $m(x)=\sqrt{x+1}$.

Answer: $m(3)=2$. Substitute $x=3$: $m(3)=\sqrt{3+1}=2$.