Functions (Advanced) - Trivium Test Prep Online Courses

### Private: Math Masters

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)

Every function has an input domain and an output range. The domain is the set of all the possible x-values that can be used as input values. The range includes all the y-values or output values that result from applying f(x). Inverse functions (written as f−1) switch the inputs and the outputs of a function. If point (a, b) is on the graph of f(x), then point (b, a) will be on the graph of f−1(x). The domain of a function will be the range of its inverse function, and vice versa. To find the inverse of a function, swap the x and y, and then solve for y.

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
$$(f\ +\ g)(x) = f(x)\ +\ g(x)$$
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). $$f(g(x)) = f(x\ +\ 5)^{2}$$ $$= x^{2}\ +\ 10x\ +\ 25$$
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