Functions (Advanced)
Functions (Advanced)
A function is a relationship between two quantities: for every value of the independent variable (usually x), there is exactly one value of the dependent variable (usually y). Functions can be thought of as a process: when a value is put in, an action (or operation) is performed and a different value comes out.
In a function, each input has exactly one output. Graphically this means the graph passes the vertical line test: anywhere a vertical line is drawn on the graph, the line intersects the curve at exactly one point.
The notation f(x)—called function notation—is often used to write functions. The input value is x and the output value (y) is f(x). The output value can be determined by substituting the given input value into the function.
Find f(2) if f(x) = x + 3.
Substitute 2 for x and solve.
f(x) = x + 3
f(2) = 2 + 3
f(2) = 5 ↔ (2, 5)
Find the inverse of \(f(x) = 5x\ +\ 5\)
Replace f(x) with y.
\(y = 5x\ +\ 5\)
Replace every x with a y and replace every y with an x.
\(x = 5y\ +\ 5\)
Solve for y.
\(y = \frac{x}{5}\ −\ 1\)
\(y = \frac{x}{5}\ −\ 1\)
Inverse graphs can be tested by taking any point on one graph and exchanging coordinates to see if that new point is on the other curve. For example, the coordinate point (5, −2) is on the graph of the function and (−2, 5) is a point on its inverse.
Compound functions take two or more functions and combine them using operations or composition. Functions can be combined using addition, subtraction, multiplication, or division.
Compound Functions | ||
|---|---|---|
Operation | Notation | Example |
Addition | \((f\ +\ g)(x) = f(x)\ +\ g(x)\) | If f(x) = x and g(x) = x – 3, find f(x) + g(x).
\(x\ +\ (x\ −\ 3)\)
\(2x\ −\ 3\) |
Subtraction | \((f\ −\ g)(x) = f(x)\ −\ g(x)\) | If f(x) = x2 and g(x) = x + 2, find f(x) − g(x).
\(x^{2}\ −\ (x\ +\ 2)\)
\(x^{2}\ −\ x\ −\ 2\) |
Multiplication | \((fg)(x) = f(x)g(x)\) | If f(x) = x + 3 and g(x) = x − 2, find f(x)g(x).
\((x\ +\ 3)(x\ −\ 2)\)
\((x^{2}\ −\ 2x\ +\ 3x\ −\ 6\)
\((x^{2}\ +\ x\ −\ 6\) |
Division | \((\frac{f}{g})(x) = \frac{f(x)}{g(x)}\)
\((note\ that\ g(x)\ \neq\ 0)\) | If f(x) = x2 + 3x and g(x) = x, find f(x)/g(x).
\(\frac{x^{2}\ +\ 3x}{x}\)
\(\frac{x(x\ +\ 3)}{x}\)
\(x\ +\ 3\) |
Composition | \((f\ \circ\ g)(x) = f(g(x))\) | If f(x) = x2 and g(x) = x + 5, find (f ∘ g)(x).
\(f(g(x)) = f(x\ +\ 5)^{2}\)
\(= x^{2}\ +\ 10x\ +\ 25\) |