Simple Functional Relationships - ISEE Upper Level Quantitative Reasoning

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

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

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

   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.

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

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

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

   $x$ 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Study these flashcards