Advanced Math / Function Notation
Function notation and transformations
Function notation asks for outputs from specified inputs or for inputs that produce specified outputs. Transformations outside $f$ change outputs; transformations inside $f$ change inputs.
Least you need to know
- $f(a)$ means substitute $a$ for the input.
- $f(x)+k$ shifts outputs by $k$.
- $f(x-k)$ shifts the graph right by $k$.
- $-f(x)$ reflects outputs over the $x$-axis.
Key notation
- f(a) — The output of function $f$ when the input is $a$
Worked example
If $g(x)=f(x-4)$ and $f(3)=10$, then $g(7)=f(3)=10$.
Common mistakes
- Do not confuse $f(x)=k$ with $f(k)$.
- Horizontal shifts inside the function use the opposite direction from the sign.
How to recognize it
- The stem gives a formula for $f$ or relates $g$ to $f$.
- The question asks for $f(a)$, solve $f(x)=k$, or interpret a transformed graph.
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