Function Notation and Transformations

Function notation f(x) means 'the output of function f when the input is x.' Evaluating f(2) means substituting x = 2 into the function. If f(x) = 3x² - x + 4, then f(2) = 3(4) - 2 + 4 = 14.

Finding x when f(x) = 5 means solving the equation 3x² - x + 4 = 5, or 3x² - x - 1 = 0.

Transformations shift or stretch the graph of a function:

• f(x) + c: shifts graph up by c units
• f(x) - c: shifts graph down by c units
• f(x + c): shifts graph left by c units
• f(x - c): shifts graph right by c units
• af(x): vertical stretch by factor a (if |a| > 1) or compression (if |a| < 1)
• f(ax): horizontal compression by factor a
• -f(x): reflects over the x-axis

Remember: horizontal shifts are counterintuitive — adding inside the function moves left, subtracting moves right.

Composition of functions f(g(x)) means apply g first, then apply f to the result. If g(x) = x + 1 and f(x) = x², then f(g(x)) = f(x+1) = (x+1)².

Note: f(g(x)) ≠ g(f(x)) in general. Composition is not commutative.

⚠️ Common mistake: confusing f(x + 2) (horizontal shift left 2) with f(x) + 2 (vertical shift up 2). The position of the constant — inside or outside the function — matters completely.

💡 Tip: To evaluate a composition like f(g(3)), first compute g(3), then plug that number into f. Work from the inside out.