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Simple Functional
Relationships

Master how to read, interpret, and evaluate functions expressed as equations, tables, and graphs: the core of ISEE Quantitative Reasoning.

ISEE Upper Level

What Is a Function?

A function is a rule that assigns exactly one output to every input. We write \( f(x) \) to mean "the output of function \( f \) when the input is \( x \)."

Input
The value you plug in: \( x \)
Rule
The operation performed: e.g., "multiply by 2, add 3"
Output
The result: \( f(x) \)

Reading Function Notation

When you see \( f(3) = 11 \), it means: when the input is 3, the output is 11. The letter \( f \) is just a name: functions can also be called \( g \), \( h \), or any letter.

3 INPUT
\( f(x) = 2x + 5 \)
\( 2(3) + 5 = 11 \)
RULE 11 OUTPUT

Three Representations

The ISEE tests functions in three forms. You must be comfortable switching between all of them.

Equation
\( f(x) = 2x + 5 \) gives the rule directly as an algebraic formula.
Table
A list of input-output pairs. Read the pattern to find the rule.
Graph
A visual curve or line. Each point \( (x, y) \) means \( f(x) = y \).

The Core Evaluation Rule

1
Identify the rule. Write down the formula for \( f(x) \).
2
Substitute. Replace \( x \) with the given input value.
3
Simplify. Follow order of operations: exponents, multiplication, then addition.
\( f(x) = x^2 + 3 \)
Rule
\( f(4) = (4)^2 + 3 \)
Substitute 19 Answer

Reading \( f(x) \) from a Graph

On a graph, \( f(a) \) is the \( y \)-value when \( x = a \). Go to \( x = a \) on the horizontal axis, then read straight up (or down) to the curve.

1 2 3 4 1 2 3 4 5 x y
\( f(3) = 3.5 \)
\( x = 3 \)
3.5

Reading \( f(x) \) from a Table

A table lists specific input-output pairs. To find \( f(a) \), locate \( a \) in the input column and read across to the output column.

\( x \) \( f(x) \)
05
18
211
314
417

\( f(x) \) vs. Finding \( x \)

The ISEE asks two types of questions. Recognizing which type you face is essential.

Find the output
"What is \( f(5) \)?"
Substitute \( x = 5 \) into the rule and simplify.
Find the input
"If \( f(x) = 20 \), what is \( x \)?"
Set the rule equal to 20 and solve for \( x \).

Step-by-Step Strategy

1
Identify the representation. Is the function given as an equation, table, or graph?
2
Determine what you need. Are you finding the output, finding the input, or finding the rule?
3
Execute carefully. Substitute, simplify, or read the value. Use parentheses to avoid sign errors.
4
Verify. Plug your answer back into the original function to check.

Example 1: Evaluate a Function

Problem: If \( g(x) = 4x - 7 \), find \( g(6) \).

1
Rule: \( g(x) = 4x - 7 \)
2
Substitute \( x = 6 \): \( g(6) = 4(6) - 7 \)
3
Simplify: \( = 24 - 7 = \) 17

Example 2: Find the Input

Problem: If \( f(x) = 2x^2 - 3 \), and \( f(a) = 29 \), find the positive value of \( a \).

1
Set up: \( 2a^2 - 3 = 29 \)
2
Add 3: \( 2a^2 = 32 \)
3
Divide by 2: \( a^2 = 16 \)
4
Square root: \( a = \pm 4 \). The positive value is \( a = 4 \).

Example 3: Discover the Rule

Problem: The table below shows values of \( h(x) \). Find \( h(10) \).

\( x \) 1 2 3 4 5
\( h(x) \) 4 7 10 13 16
1
Find the pattern: Each output increases by 3. This is a linear function.
2
Find the rule: Slope \( = 3 \). When \( x = 1 \), \( h = 4 \), so \( h(x) = 3x + 1 \).
3
Evaluate: \( h(10) = 3(10) + 1 = \) 31.

Example 4: Nested Evaluation

Problem: If \( f(x) = x + 3 \) and \( g(x) = 2x \), find \( f(g(4)) \).

4
\( g(x) = 2x \)
\( g(4) = 8 \)
\( f(x) = x + 3 \)
\( f(8) = 11 \)
11 INPUT OUT

Common Mistakes

✕ Treating \( f(x) \) as multiplication
\( f(3) \neq f \times 3 \). The parentheses mean "evaluate at," not "multiply."
✕ Forgetting parentheses
When squaring a negative: \( f(-2) = (-2)^2 = 4 \), not \( -4 \). Always wrap the substitution in parentheses.
✕ Mixing up input and output
"\( f(x) = 10 \)" asks for \( x \), not \( f(10) \). Read the question carefully.

Quick Reference Summary

Task From Equation From Table From Graph
Find \( f(a) \) Substitute \( x = a \) Look up row \( x = a \) Read \( y \) at \( x = a \)
Find \( x \) when \( f(x) = k \) Solve equation for \( x \) Look up row \( f(x) = k \) Read \( x \) at \( y = k \)
Find the rule Already given Find the pattern Identify shape and slope

Question 1 of 5

If \( f(x) = 3x - 5 \), what is \( f(7) \)?

Question 2 of 5

If \( g(x) = x^2 + 2x \), what is \( g(-3) \)?

Question 3 of 5

The table shows values of \( h(x) \). If the pattern continues, what is \( h(8) \)?

\( x \) 1234
\( h(x) \) 591317

Question 4 of 5

If \( f(x) = 5x + 3 \) and \( f(a) = 28 \), what is the value of \( a \)?

Question 5 of 5

If \( f(x) = x + 4 \) and \( g(x) = 3x \), what is \( g(f(2)) \)?