GRE Quantitative Flashcards: Functions Function Notation

What this deck covers

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

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 the notation $f(x)$ represent in function notation?

Answer: $f(x)$ is the output value of function $f$ for input $x$. This notation evaluates the function (f) at the specific input value (x), yielding the corresponding output.

Flashcard 2: What is the domain of $f(x)=\sqrt{x+1}$ in interval notation?

Answer: $[-1,\infty)$. Require the expression inside the square root to be non-negative to yield real outputs.

Flashcard 3: What is the domain of $f(x)=\sqrt{5-x}$ in interval notation?

Answer: $(-\infty,5]$. Ensure the radicand is non-negative for the square root function to be defined over reals.

Flashcard 4: What is the meaning of the statement $f(a)=b$?

Answer: Input $a$ maps to output $b$ under $f$. The equation states that applying function (f) to input (a) produces the output value (b).

Flashcard 5: What is $g(f(x))$ if $f(x)=x^2$ and $g(x)=x-1$?

Answer: $g(f(x))=x^2-1$. Input (f(x)) into (g), resulting in subtraction after squaring.

Flashcard 6: What is $(g\circ f)(2)$ if $f(x)=x+3$ and $g(x)=2x$?

Answer: $(g\circ f)(2)=10$. Evaluate (f(2)) first, then apply (g) to that output for the composition.

Flashcard 7: What is the key distinction between $f^{-1}(x)$ and $\frac{1}{f(x)}$?

Answer: $f^{-1}(x)$ is inverse; $\frac{1}{f(x)}$ is reciprocal. The inverse undoes the function's operation, while the reciprocal is the multiplicative inverse of the output.

Flashcard 8: What is the range of a function (in words) in function notation context?

Answer: The set of all possible output values $f(x)$. The range includes every possible output obtained by applying the function to values in its domain.

Flashcard 9: What is the value of $f(1)$ if $f(x)=\begin{cases}x+1,&x<2\\3x,&x\ge 2\end{cases}$?

Answer: $f(1)=2$. As (1 < 2), apply the first piece of the piecewise definition for computation.

Flashcard 10: What is the meaning of $f(x)=f(y)$ for a one-to-one function $f$?

Answer: It implies $x=y$. One-to-one functions are injective, so equal outputs necessitate equal inputs.

Flashcard 11: What is the value of $f(3)-f(1)$ if $f(x)=x^2+2$?

Answer: $f(3)-f(1)=8$. Compute each function value separately using the quadratic formula, then subtract.

Flashcard 12: What is the value of $f(2)$ if $f(x)=\begin{cases}x+1,&x<2\\3x,&x\ge 2\end{cases}$?

Answer: $f(2)=6$. Since (2 geq 2), use the second piece of the piecewise function to evaluate.

Flashcard 13: What is $(f\circ g)(2)$ if $f(x)=x+3$ and $g(x)=2x$?

Answer: $(f\circ g)(2)=7$. First compute (g(2)), then input that result into (f) to find the composed value.

Flashcard 14: What is the definition of an inverse function in function notation?

Answer: $f^{-1}(y)=x$ exactly when $f(x)=y$. The inverse function reverses the mapping, so inputs and outputs are swapped.

Flashcard 15: What is $f(x-3)$ if $f(x)=x^2$?

Answer: $f(x-3)=(x-3)^2$. Substitute the expression (x-3) into the quadratic function in place of (x).

Flashcard 16: What is the composition notation $(f\circ g)(x)$ equal to?

Answer: $(f\circ g)(x)=f(g(x))$. Function composition applies (f) to the result of (g(x)), nesting the functions.

Flashcard 17: What is the domain of a function (in words) in function notation context?

Answer: The set of all allowed input values $x$. The domain consists of every input (x) for which the function is defined and produces a valid output.

Flashcard 18: What is the domain of $f(x)=\frac{1}{x-5}$ in interval notation?

Answer: $(-\infty,5)\cup(5,\infty)$. Exclude the value where the denominator is zero to ensure the function is defined for all real inputs.

Flashcard 19: What is $f(g(x))$ if $f(x)=x^2$ and $g(x)=x-1$?

Answer: $f(g(x))=(x-1)^2$. Substitute (g(x)) into (f) as its argument, yielding the squared expression.

Flashcard 20: What is the value of $f(-2)$ if $f(x)=3x-5$?

Answer: $f(-2)=-11$. Substitute (x = -2) into the linear function to compute the output directly.

Flashcard 21: What is $f^{-1}(9)$ if $f(x)=x^2$ with domain restricted to $x\ge 0$?

Answer: $f^{-1}(9)=3$. With the domain restriction ensuring one-to-one, solve ($x^2$ = 9) and take the non-negative root.

Flashcard 22: What is the value of $g(4)$ if $g(x)=x^2-1$?

Answer: $g(4)=15$. Evaluate the quadratic function by plugging in (x = 4) and simplifying.

Flashcard 23: What is the value of $f(a+1)$ if $f(x)=2x-7$?

Answer: $f(a+1)=2a-5$. Replace (x) with (a+1) in the linear expression and simplify algebraically.

Flashcard 24: What is the value of $h(0)$ if $h(x)=\frac{x+2}{3}$?

Answer: $h(0)=\frac{2}{3}$. Insert (x = 0) into the rational function and reduce the fraction.