📝 Functions Graphs

Part 1 of 7 — Function Notation & Evaluation

f(x) means "the output of function f when the input is x".

To evaluate f(3): substitute 3 everywhere you see x.

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Key Insight: f(a) = 0 means a is a zero (x-intercept) of the function.

SAT Tip: If f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7.

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Domain and Range

Part 2 of 7 — Domain and Range

Domain: all possible input (x) values.

Range: all possible output (y) values.

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Key Insight: Restrictions: no division by zero, no square root of negatives (for reals).

SAT Tip: From a graph: domain is the horizontal extent, range is the vertical extent.

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Interpreting Graphs

Part 3 of 7 — Interpreting Graphs

Increasing: graph goes up left to right; decreasing: goes down.

Maximum/minimum: highest/lowest point on the graph.

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Key Insight: Intercepts: where the graph crosses the axes.

SAT Tip: Rate of change = (change in y)/(change in x) = slope between two points.

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Transformations of Functions

Part 4 of 7 — Transformations of Functions

f(x) + k: shifts up k units; f(x) - k: shifts down k.

f(x - h): shifts right h units; f(x + h): shifts left h.

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Key Insight: af(x): vertical stretch (a > 1) or compression (0 < a < 1).

SAT Tip: f(-x): reflects over y-axis; -f(x): reflects over x-axis.

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Function Composition

Part 5 of 7 — Function Composition

(f ∘ g)(x) = f(g(x)): apply g first, then f.

Find g(x) first, then use that result as input to f.

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Key Insight: Domain of f ∘ g: x must be in domain of g, AND g(x) must be in domain of f.

SAT Tip: Example: f(x) = x², g(x) = x + 1, then f(g(x)) = (x+1)².

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Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

(f ∘ g)(x) = f(g(x)): apply g first, then f.

Find g(x) first, then use that result as input to f.

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Key Insight: Domain of f ∘ g: x must be in domain of g, AND g(x) must be in domain of f.

SAT Tip: Example: f(x) = x², g(x) = x + 1, then f(g(x)) = (x+1)².

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Review & Applications

Part 7 of 7 — Review & Applications

(f ∘ g)(x) = f(g(x)): apply g first, then f.

Find g(x) first, then use that result as input to f.

Check Your Understanding 🎯

Key Insight: Domain of f ∘ g: x must be in domain of g, AND g(x) must be in domain of f.

SAT Tip: Example: f(x) = x², g(x) = x + 1, then f(g(x)) = (x+1)².

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Match the Concepts 🔍