ISEE Middle Level

Functions

Q: What does f(x) actually mean?

It's just a fancy math way of writing `y`! It tells us that the output (the answer) depends on the input (the `x`). You say it out loud as 'f of x'.

Q: Do I need to memorize the slope formula for the ISEE?

Yes! The ISEE doesn't give you a formula sheet. Memorize `m = \frac{y_2 - y_1}{x_2 - x_1}` and remember to always put the `y` values on top!

Function notation f(x), evaluating functions, input/output tables, domain, and range — does NOT include graphing

Generate Unlimited Practice Questions

Start Free Practice

Learn This Topic

Have you ever used a vending machine? You push a button (the input), and out pops a snack (the output). In math, a function is just like a magical smoothie machine! 🍓 You drop a number in, the machine chops it up, mixes it with some math rules, and pours out a brand new number.

When you see something like $f (x) = 2 x + 1$ , don't let those letters scare you! The $f$ just stands for 'function', and the $x$ is the ingredient you drop inside. If you drop in a $3$ (we write that as $f (3)$ ), the machine multiplies it by $2$ and adds $1$ to give you $7$ . Delicious! 🥤

Graphing is just a way to take a picture of what our machine is doing. We use a coordinate plane, which looks like a giant piece of grid paper. Every time we put an $x$ into our machine and get a $y$ out, we get a point (x, y). If we plot a bunch of points and connect them, we get a line!

On the ISEE, you'll see questions asking you to find points on a line, figure out the slope (how steep the line is, like a skateboard ramp! 🛹), or compare functions in the Quantitative Reasoning section. The great news? The ISEE has no penalty for guessing! So, always pick an answer, even if the math machine seems a little tricky that day. Grab your pencil and let's make some math smoothies!

Key Formula

The Magic Line Equation (Slope-Intercept Form) is

y = m x + b

. The

m

is the slope (how steep the line is) and

b

is the y-intercept (where the line crosses the y-axis). To find the slope between two points, use

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Practice Questions

4 practice questions for ISEE Middle Level

Q1 Medium

f (x) = 3 x^{2} - 2 x + 5

, what is the value of

f (- 2)

A 13

B 21

C 17

D 9

Show Solution

To find $f (- 2)$ , substitute $x = - 2$ into the function $f (x) = 3 x^{2} - 2 x + 5$ .

$f (- 2) = 3 (- 2)^{2} - 2 (- 2) + 5$
First, calculate $(- 2)^{2}$ : $(- 2)^{2} = 4$ .
$f (- 2) = 3 (4) - 2 (- 2) + 5$
Next, perform the multiplications: $3 (4) = 12$ and $2 (- 2) = - 4$ .
$f (- 2) = 12 - (- 4) + 5$
Simplify $- (- 4)$ to $+ 4$ :
$f (- 2) = 12 + 4 + 5$
Finally, perform the additions:
$f (- 2) = 16 + 5$
$f (- 2) = 21$

Answer: B

Q2 Medium

The table below shows a set of input (

x

) and output (

y

) values for a function.

$x$	$y$
1	5
2	8
3	11
4	14

Which of the following functions represents the relationship between

x

and

y

f (x) = 2 x + 3

f (x) = 3 x + 2

f (x) = x + 4

f (x) = 4 x + 1

Show Solution

We need to find a function $f (x)$ that, when $x$ is substituted, produces the corresponding $y$ value in the table. Let's test each choice:

For Choice A: $f (x) = 2 x + 3$
- If $x = 1$ , $f (1) = 2 (1) + 3 = 2 + 3 = 5$ (Matches)
- If $x = 2$ , $f (2) = 2 (2) + 3 = 4 + 3 = 7$ (Does NOT match $y = 8$ )
So, Choice A is incorrect.
For Choice B: $f (x) = 3 x + 2$
- If $x = 1$ , $f (1) = 3 (1) + 2 = 3 + 2 = 5$ (Matches)
- If $x = 2$ , $f (2) = 3 (2) + 2 = 6 + 2 = 8$ (Matches)
- If $x = 3$ , $f (3) = 3 (3) + 2 = 9 + 2 = 11$ (Matches)
- If $x = 4$ , $f (4) = 3 (4) + 2 = 12 + 2 = 14$ (Matches)
All values match, so Choice B is the correct function.

Answer: B

Q3 Medium

A local gym charges a one-time registration fee of $30 and a monthly membership fee of $45. The total cost,

C (m)

, for

m

months of membership can be represented by a function. What is the total cost for 7 months of membership?

A $315

B $345

C $360

D $390

Show Solution

First, let's write the function for the total cost, $C (m)$ , where $m$ is the number of months.

The gym charges a one-time registration fee of $30. This is a fixed cost, independent of the number of months.
The monthly membership fee is $45. This means for $m$ months, the cost will be $45 m$ .
So, the total cost function is $C (m) = 45 m + 30$ .
To find the total cost for 7 months of membership, we need to evaluate $C (7)$ .
Substitute $m = 7$ into the function:
$C (7) = 45 (7) + 30$
First, calculate the multiplication: $45 \times 7$ .
$45 \times 7 = 315$
Now, add the registration fee:
$C (7) = 315 + 30$
$C (7) = 345$
The total cost for 7 months of membership is $345.

Answer: B

Q4 Medium

For the function defined by the set of ordered pairs

(1, 5), (2, 7), (3, 5), (4, 9)

, what is the range of the function?

1, 2, 3, 4

5, 7, 9

1, 2, 3, 4, 5, 7, 9

5, 7, 5, 9

Show Solution

The domain of a function is the set of all possible input values (x-coordinates).

The range of a function is the set of all possible output values (y-coordinates).
Given the set of ordered pairs: $(1, 5), (2, 7), (3, 5), (4, 9)$
The x-coordinates (inputs) are 1, 2, 3, 4. So the domain is $1, 2, 3, 4$ .
The y-coordinates (outputs) are 5, 7, 5, 9.
To find the range, we list all the unique y-coordinates. We do not list the value $5$ twice.
The unique y-coordinates are 5, 7, 9.
Therefore, the range of the function is $5, 7, 9$ .

Answer: B

Tips & Strategies

Rise over Run! When finding slope on a graph, always count how far UP or DOWN you go (rise) before you count how far LEFT or RIGHT you go (run). Putting the $x$ on top of your fraction is a super common trap!
For Quantitative Comparison questions, don't solve for $x$ or $y$ if you don't have to! If they just ask for the y-intercept, grab the $b$ from $y = m x + b$ and compare. Save your brainpower for harder questions!
Use the answers! If you forget how to graph a line, you can test the points given in the answer choices by plugging the $x$ and $y$ values back into the equation to see if it makes a true math sentence.

Common Mistakes

Watch out for negative signs when finding the slope! If you are subtracting a negative number, like $5 - (- 2)$ , remember that two negatives make a positive, so it becomes $5 + 2$ .
Don't get tricked by $f (x)$ . It does NOT mean $f$ times $x$ ! It is just the name of the function, like saying 'Machine A'.

Frequently Asked Questions

What does f(x) actually mean?

It's just a fancy math way of writing $y$ ! It tells us that the output (the answer) depends on the input (the $x$ ). You say it out loud as 'f of x'.

Do I need to memorize the slope formula for the ISEE?

Yes! The ISEE doesn't give you a formula sheet. Memorize $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ and remember to always put the $y$ values on top!

What if I get totally stuck on a function question?

Since the ISEE has no guessing penalty, never leave a question blank! If you're stuck, try plugging in simple numbers like $0$ or $1$ for $x$ to see what happens, or just make your best guess and move on.

Generate Unlimited Practice Questions

Learn This Topic

Practice Questions

Tips & Strategies

Common Mistakes

Frequently Asked Questions

Related Topics

More Algebra

Other Levels

Browse

Generate Unlimited Practice Questions