🎯 Learning Objectives
- How to read and use function notation like f(x), g(x), and f(g(x)).
- How to evaluate a function at a number or another expression.
- How shifts, stretches, and reflections change a graph.
- Domain and range: what x and y values are allowed.
- Strategies for composite functions.
- Practice questions with worked solutions.
1. What Is Function Notation?
f(x) means 'apply the function f to the input x.' The notation is just a fancy way of saying 'plug x into the rule.' f(2) means substitute 2 wherever you see x in the rule.
Plug-and-chug example
If f(x) = 3x + 1, then f(5) = 3(5) + 1 = 16
2. Evaluating Functions at Expressions
You can plug in not just numbers but entire expressions. Wherever you see x in the rule, write the expression in parentheses.
📝 Example: If f(x) = x² − 4x, find f(a + 1)
Replace x with (a + 1): f(a+1) = (a+1)² − 4(a+1)
Expand: (a² + 2a + 1) − (4a + 4) = a² − 2a − 3
3. Composite Functions: f(g(x))
f(g(x)) means: 'first do g, then plug that result into f.' Work from the inside out.
📝 Example: f(x) = x + 5, g(x) = 2x. Find f(g(3)).
Inside first: g(3) = 2(3) = 6
Now apply f: f(6) = 6 + 5 = 11
4. The 4 Transformations
Change | Effect on Graph | Direction |
f(x) + k | Vertical shift | Up if k > 0; down if k < 0 |
f(x − h) | Horizontal shift | RIGHT if h > 0; LEFT if h < 0 |
a · f(x) | Vertical stretch / shrink | Stretch if |a| > 1; shrink if |a| < 1 |
−f(x) | Reflection | Flips across the x-axis |
f(−x) | Reflection | Flips across the y-axis |
💡 The Counterintuitive Rule
Inside-the-parentheses changes do the OPPOSITE of what they look like:
- f(x − 3) shifts RIGHT 3 (not left)
- f(x + 5) shifts LEFT 5 (not right)
Outside changes work normally: f(x) + 4 shifts UP 4.
5. Domain & Range
Concept | Definition | Common Restrictions |
Domain | All allowed x-values (inputs) | No /0; no √(negative); etc. |
Range | All possible y-values (outputs) | Depends on the function shape. |
6. Strategies
⚡ STRATEGY 1: Plug fast for evaluation
When asked for f(7), don't simplify f(x) algebraically: just substitute 7 directly into the formula.
⚡ STRATEGY 2: Composite from a TABLE
When given a table of f and g values, find f(g(3)) by reading g(3) from the table, then looking up that result in f's column.
⚡ STRATEGY 3: Spot the transformation
Look at the equation: anything OUTSIDE acts vertically; anything INSIDE acts horizontally (and is reversed in direction).
⚠️ Common Mistakes
- Reading f(2) as 'f times 2': it's 'f at 2.'
- Confusing f(g(x)) with g(f(x)): order matters!
- Shifting LEFT for f(x − 3) instead of RIGHT.
- Forgetting parentheses when substituting an expression: f(a + 1) ≠ f(a) + 1.
7. Summary
📌 Key Takeaways
- f(x): 'plug x into the rule.'
- f(g(x)): Inside out: first g, then f.
- f(x) + k: Vertical shift (k > 0 → up).
- f(x − h): Horizontal shift RIGHT by h (counterintuitive!).
- −f(x): Reflect across x-axis.
- f(−x): Reflect across y-axis.
- Domain: Allowed x. Range: possible y.
8. Practice: 10 Questions
Q1. If f(x) = 2x + 7, what is f(4)?
A) 11 B) 15 C) 18 D) 24
Hint: Plug 4: 8 + 7 = 15.
Q2. If f(x) = x² − x, what is f(−2)?
A) 2 B) 4 C) 6 D) 8
Hint: (−2)² − (−2) = 4 + 2 = 6.
Q3. If f(x) = 3x and g(x) = x + 1, what is f(g(2))?
A) 6 B) 7 C) 8 D) 9
Hint: g(2) = 3, f(3) = 9.
Q4. The graph of y = x² is shifted RIGHT 2. The new equation is…
A) y = (x + 2)² B) y = (x − 2)² C) y = x² + 2 D) y = x² − 2
Hint: Inside, opposite: right means (x − 2).
Q5. Which transformation moves y = f(x) DOWN 5?
A) y = f(x − 5) B) y = f(x + 5) C) y = f(x) + 5 D) y = f(x) − 5
Hint: Outside subtraction = down.
Q6. The domain of f(x) = √(x − 4) is…
A) x > 0 B) x ≥ 4 C) x ≤ 4 D) all real x
Hint: Need x − 4 ≥ 0 → x ≥ 4.
Q7. If f(x) = x + 3 and g(x) = 2x, then g(f(x)) = ?
A) 2x + 3 B) 2x + 6 C) x + 6 D) 3x + 6
Hint: f first: x + 3. Then g: 2(x + 3) = 2x + 6.
Q8. The graph of y = f(x) is reflected across the x-axis. The new equation is…
A) y = f(−x) B) y = −f(x) C) y = f(x) − 1 D) y = 1/f(x)
Hint: Negative outside = x-axis reflection.
Q9. If f(x) = x² + 1, what is f(a − 2)?
A) a² + 1 B) (a−2)² + 1 C) a² − 4a + 5 D) Both B and C
Hint: Plug (a−2): (a−2)² + 1 = a² − 4a + 5.
Q10. If f(2) = 7, what is the value of g(x) = f(x) + 3 at x = 2?
A) 7 B) 10 C) 14 D) 21
Hint: g(2) = f(2) + 3 = 7 + 3 = 10.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | B) 15 | Direct plug. |
Q2. | C) 6 | Watch the sign. |
Q3. | D) 9 | g(2) = 3, then f(3) = 9. |
Q4. | B) y = (x − 2)² | Inside: opposite direction. |
Q5. | D) y = f(x) − 5 | Outside subtraction = down. |
Q6. | B) x ≥ 4 | Radicand ≥ 0. |
Q7. | B) 2x + 6 | Apply f first, then g. |
Q8. | B) y = −f(x) | Negative outside. |
Q9. | D) Both B and C | Equivalent forms. |
Q10. | B) 10 | g(2) = f(2) + 3. |