SSAT Upper 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

7 practice questions for SSAT Upper Level

Q1 Hard

Let the function

f

be defined as

f (x) = 5 x - 4

. For what value of

b

f (b)

equal to 21?

A 3

B 4

C 5

D 21

E 101

Show Solution

Substitute $b$ into the function to get $f (b) = 5 b - 4$ . Set this expression equal to 21: $5 b - 4 = 21$ . Add 4 to both sides of the equation to get $5 b = 25$ . Finally, divide by 5 to find $b = 5$ .

Answer: C

Q2 Hard

The function

g

is defined by the values in the table below. What is the value of

g (- 1) \times g (2)

$x$	-2	-1	0	1	2
$g (x)$	5	2	-1	3	7

A -2

B 5

C 9

D 14

E 15

Show Solution

Use the table to find the outputs of the function for the given $x$ values. When $x = - 1$ , the value of $g (x)$ is 2, so $g (- 1) = 2$ . When $x = 2$ , the value of $g (x)$ is 7, so $g (2) = 7$ . Multiply these two values together: $2 \times 7 = 14$ .

Answer: D

Q3 Hard

Let the functions

h

and

p

be defined as

h (x) = x^{2} - 2 x

and

p (x) = 10 - 2 x

. If

h (- 3) = c

, what is the value of

p (c)

A -20

B -10

C 4

D 15

E 40

Show Solution

First, find the value of $c$ by evaluating $h (- 3)$ . Substitute $- 3$ into the function $h$ : $h (- 3) = (- 3)^{2} - 2 (- 3) = 9 - (- 6) = 9 + 6 = 15$ . Thus, $c = 15$ . Next, find the value of $p (c)$ , which is $p (15)$ . Substitute 15 into the function $p$ : $p (15) = 10 - 2 (15) = 10 - 30 = - 20$ .

Answer: A

Q4 Hard

Let the operation

△

be defined for all real numbers

x

and

y

by the equation

x △ y = \frac{x ^{2} - y}{2}

. What is the value of

4△ (- 6)

A 5

B 8

C 11

D 14

E 22

Show Solution

Substitute $x = 4$ and $y = - 6$ into the given equation: $4△ (- 6) = \frac{4 ^{2} - ( - 6 )}{2}$ . First, evaluate the exponent: $4^{2} = 16$ . Next, subtract negative 6, which is equivalent to adding 6: $16 - (- 6) = 16 + 6 = 22$ . Finally, divide by 2: $\frac{22}{2} = 11$ .

Answer: C

Q5 Hard

Let the operation

⋆

be defined for all integers

x

and

y

x ⋆ y = 2 x - y

. What is the value of

5 ⋆ 3

A 13

B 10

C 7

D 4

E 2

Show Solution

To find the value of $5 ⋆ 3$ , substitute $x = 5$ and $y = 3$ into the given expression $2 x - y$ . This gives $2 (5) - 3 = 10 - 3 = 7$ .

Answer: C

Q6 Hard

For any two numbers

a

and

b

, the operation

⋄

is defined as

a ⋄ b = \frac{a + b}{2}

. What is the value of

(6 ⋄ 10) ⋄ 4

A 10

B 8

C 6

D 5

E 4

Show Solution

First, follow the order of operations and calculate the value inside the parentheses: $6 ⋄ 10$ . Substitute $a = 6$ and $b = 10$ into the expression to get $\frac{6 + 10}{2} = \frac{16}{2} = 8$ . Now, substitute this result back into the remaining expression to find $8 ⋄ 4$ . Using the rule again, $\frac{8 + 4}{2} = \frac{12}{2} = 6$ .

Answer: C

Q7 Hard

Let

△ x

be defined as

△ x = x^{2} - x

for all integers

x

. What is the value of

△4

A 20

B 16

C 12

D 8

E 4

Show Solution

To find the value of $△4$ , substitute $x = 4$ into the defined expression $x^{2} - x$ . This gives $4^{2} - 4$ . First, calculate the exponent: $4^{2} = 16$ . Then subtract 4: $16 - 4 = 12$ .

Answer: C

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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