SSAT Middle Level

Functions

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

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Imagine a magical machine in a factory. You drop in a plain piece of dough, the machine buzzes and whirs, and out pops a delicious, hot pizza! 🍕 That is exactly how a mathematical function works. It is simply a rule machine. You put a number in (the input, usually called $x$ ), the machine does some math to it, and it spits out a brand new number (the output, usually called $y$ or $f (x)$ ).

If the machine's rule is 'add 5,' and you put in a 2, out comes a 7. Easy, right?

Now, imagine taking all those inputs and outputs and playing a giant game of Connect the Dots. That is what graphing is! 📈 We use a coordinate plane, which is just a flat grid. The x-axis goes left to right like a flat road, and the y-axis goes up and down like a tall building. Every point on the graph is a special address written as (x, y). This address shows exactly what went into the machine ( $x$ ) and what came out ( $y$ ).

On the SSAT, you might be asked to plug numbers into a function machine, find the slope (how steep a line is), or figure out where a line lives on a graph. Don't worry if it looks like a secret code at first. Once you learn the basic rules of the machine, you will be a function master in no time! 🦸‍♂️

Key Formula

To find the slope (steepness) of a line between two points, use the 'Rise over Run' formula:

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

Practice Questions

4 practice questions for SSAT Middle Level

Q1 Medium

f (x) = 2 x^{2} - 3 x + 1

, what is the value of

f (4)

A 17

B 21

C 24

D 29

E 33

Show Solution

To find $f (4)$ , substitute $x = 4$ into the function $f (x)$ .

$f (4) = 2 (4)^{2} - 3 (4) + 1$
First, calculate the exponent:
$f (4) = 2 (16) - 3 (4) + 1$
Then, perform the multiplications:
$f (4) = 32 - 12 + 1$
Finally, perform the additions and subtractions:
$f (4) = 20 + 1$
$f (4) = 21$

Answer: B

Q2 Medium

A car rental company charges a daily fee based on the number of days a car is rented. The cost

C (d)

in dollars for renting a car for

d

days is given by the function

C (d) = 35 d + 20

. What is the cost of renting a car for 5 days?

A $

175

B $

195

C $

205

D $

215

E $

225

Show Solution

The function $C (d) = 35 d + 20$ describes the cost of renting a car for $d$ days.

To find the cost for 5 days, substitute $d = 5$ into the function:
$C (5) = 35 (5) + 20$
Perform the multiplication:
$C (5) = 175 + 20$
Perform the addition:
$C (5) = 195$
So, the cost of renting a car for 5 days is $ $195$ .

Answer: B

Q3 Medium

Consider the function

g (x) = x + 7

, where the domain is

- 2, 0, 1, 5

. What is the range of

g (x)

- 9, - 7, - 6, - 2

- 2, 0, 1, 5

5, 7, 8, 12

- 2, 5, 7, 12

0, 1, 5, 7

Show Solution

The domain is the set of all possible input values for $x$ . To find the range, we substitute each value from the domain into the function $g (x) = x + 7$ :

For $x = - 2$ : $g (- 2) = - 2 + 7 = 5$
For $x = 0$ : $g (0) = 0 + 7 = 7$
For $x = 1$ : $g (1) = 1 + 7 = 8$
For $x = 5$ : $g (5) = 5 + 7 = 12$
The range is the set of all output values, so the range is $5, 7, 8, 12$ .

Answer: C

Q4 Medium

h (t) = t^{2} + 5

and

t = 2 p - 1

, what is

h (t)

in terms of

p

4 p^{2} + 4 p + 6

4 p^{2} - 4 p + 6

4 p^{2} + 6

2 p^{2} + 4

4 p^{2} - 4 p + 4

Show Solution

We are given two expressions: $h (t) = t^{2} + 5$ and $t = 2 p - 1$ .

To find $h (t)$ in terms of $p$ , we substitute the expression for $t$ into the function $h (t)$ :
$h (t) = (2 p - 1)^{2} + 5$
First, expand $(2 p - 1)^{2}$ using the formula $(a - b)^{2} = a^{2} - 2 ab + b^{2}$ or by multiplying $(2 p - 1) (2 p - 1)$ :
$(2 p - 1)^{2} = (2 p)^{2} - 2 (2 p) (1) + 1^{2}$
$= 4 p^{2} - 4 p + 1$
Now substitute this expanded expression back into $h (t)$ :
$h (t) = (4 p^{2} - 4 p + 1) + 5$
Combine the constant terms:
$h (t) = 4 p^{2} - 4 p + 6$

Answer: B

Tips & Strategies

When you see $f (x)$ , don't get scared! It is just a fancy way of writing $y$ . Treat it like a set of instructions telling you exactly what number to plug into the $x$ spot. 🤖
If a question asks which point is on a line, just plug the $x$ and $y$ values from the answer choices into the equation. The one that makes the math perfectly true is your winner! 🏆
Always physically write labels like $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ above your points before using the slope formula. It takes two extra seconds and prevents silly mix-ups!

Common Mistakes

Watch out for negative numbers when using the slope formula! If you subtract a negative number, it turns into a positive (for example, $5 - (- 2) = 5 + 2 = 7$ ). ⚠️
Don't flip your x and y axis! Remember that $x$ means going left or right (walking across the floor), and $y$ means going up or down (climbing the stairs). 🚪🚶‍♂️

Frequently Asked Questions

What does 'f(x)' even mean?

It is read out loud as 'f of x'. It just tells you the name of the function is 'f' and the input variable is 'x'. It's exactly the same as 'y' in a regular equation!

How do I remember which axis is which?

Think of a real-life cross. The x-axis is a flat floor you walk aCross (horizontal). The y-axis is a tall tree with branches pointing up to the skY (vertical). 🌳

Will I have to draw graphs on the SSAT?

Nope! The SSAT is entirely multiple choice. You won't have to draw your own graphs, but you will need to read them, find points on them, or calculate slopes from points.

What if my slope fraction is upside down?

That's a super common error! Always remember 'Rise over Run.' The Y values (up and down) must ALWAYS go on top of the fraction, like $\frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ .

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