ISEE Upper Level Quantitative Reasoning Flashcards: Simple Functional Relationships

What this deck covers

This deck focuses on Simple Functional Relationships, giving you a quick way to review the definitions, rules, and examples that matter most for ISEE Upper Level Quantitative Reasoning.

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 is the input $x$ if $f(x)=2x+7$ and $f(x)=15$?

Answer: $4$. Solve the equation $2x + 7 = 15$ by subtracting 7 and dividing by 2 to find the input.

Flashcard 2: What is the definition of a function in terms of inputs and outputs?

Answer: A relation where each input has exactly one output. A function ensures that for every input value in the domain, there is precisely one corresponding output value.

Flashcard 3: What is the average rate of change of $f$ from $x=a$ to $x=b$?

Answer: $\frac{f(b)-f(a)}{b-a}$. The average rate of change is the slope of the secant line between $x=a$ and $x=b$.

Flashcard 4: What do $x$ and $y$ represent in a functional relationship $y=f(x)$?

Answer: $x$ is the input; $y$ is the output. In the notation $y = f(x)$, $x$ serves as the independent variable or input, while $y$ is the dependent variable or output.

Flashcard 5: What is the function rule if a table shows $(1,4)$ and $(2,7)$ and the relationship is linear?

Answer: $f(x)=3x+1$. Find the slope from the points, then use point-slope form to derive the linear equation.

Flashcard 6: What is the value of $f(2x)$ if $f(x)=3x-1$ (simplify)?

Answer: $6x-1$. Replace $x$ with $2x$ in the linear function and distribute the coefficient.

Flashcard 7: What is the domain restriction for $f(x)=\sqrt{x+5}$?

Answer: $x\ge -5$. For real square roots, the expression inside must be non-negative, setting the domain boundary.

Flashcard 8: What is $k$ if $y$ is proportional to $x$ and $(x,y)=(4,10)$?

Answer: $\frac{5}{2}$. Determine $k$ by dividing $y$ by $x$ from the given point in the proportional relationship.

Flashcard 9: What is the value of $f(x+2)$ if $f(x)=x^2$ (simplify)?

Answer: $x^2+4x+4$. Substitute $x+2$ into the function and expand the binomial to simplify.

Flashcard 10: What is the function rule for a proportional relationship with constant of proportionality $k$?

Answer: $y=kx$. A proportional relationship passes through the origin with slope equal to the constant $k$.

Flashcard 11: What is the average rate of change for $f(x)=x^2$ from $x=1$ to $x=4$?

Answer: $5$. Compute using the formula for average rate of change over the interval from 1 to 4.

Flashcard 12: What is the inverse of the function $f(x)=x+7$?

Answer: $f^{-1}(x)=x-7$. The inverse undoes the operation by subtracting 7 to recover the original input.

Flashcard 13: What is the output when $f(x)=3x-2$ and $x=5$?

Answer: $13$. Substitute $x=5$ into the linear function to compute the output as $3(5) - 2$.

Flashcard 14: What is $f(-2)$ if $f(x)=x^2+1$?

Answer: $5$. Evaluate the quadratic function by substituting $x = -2$ to get $(-2)^2 + 1$.

Flashcard 15: What is the range of $f(x)=x^2-4$ for all real $x$?

Answer: $y\ge -4$. The parabola shifts down by 4 units, making the minimum value -4 at the vertex.

Flashcard 16: What is the domain restriction for $f(x)=\frac{1}{x-3}$?

Answer: $x\ne 3$. The domain excludes values that make the denominator zero to avoid undefined results.

Flashcard 17: Which ordered pair correctly represents $f(3)$ if $f(3)=-2$?

Answer: $(3,-2)$. The ordered pair lists the input first and the corresponding output second.

Flashcard 18: What is the $y$-intercept of $y=-4x+9$?

Answer: $9$. In slope-intercept form, the constant term is the $y$-coordinate where the line crosses the $y$-axis.

Flashcard 19: What is the output if $f(x)=|x-5|$ and $x=2$?

Answer: $3$. The absolute value function computes the distance from $x$ to 5, yielding a non-negative result.

Flashcard 20: What is the slope of the line through $(2,5)$ and $(6,13)$?

Answer: $2$. Calculate the slope using the formula $\frac{y_2 - y_1}{x_2 - x_1}$ with the given points.

Flashcard 21: What is the slope of the linear function $y=mx+b$ in terms of change in $x$ and $y$?

Answer: $m=\frac{\Delta y}{\Delta x}$. The slope $m$ represents the ratio of the change in $y$ to the change in $x$ for a linear function.

Flashcard 22: What is the output if $f(x)=2^x$ and $x=3$?

Answer: $8$. Evaluate the exponential function by raising the base 2 to the power of 3.

Flashcard 23: What is the $y$-intercept of a line given by $y=mx+b$?

Answer: The value of $b$. The $y$-intercept occurs when $x=0$, so it equals $b$ in the slope-intercept form.

Flashcard 24: What is the range of the function $f(x)=x^2$ for all real $x$?

Answer: $y\ge 0$. The square of any real number is non-negative, so outputs are bounded below by zero.

Flashcard 25: What is $f(g(2))$ if $f(x)=x+3$ and $g(x)=2x$?

Answer: $7$. First evaluate $g(2)$, then substitute that result into $f$ for composition.