The best way to begin analyzing a function is to identify what type of function it is, its category.

This is especially important in Calculus.

In Calculus, you will be applying the rules of differentiation to functions. That means selecting a derivative rule. Each of the derivative rules corresponds to a type of expression.

If you cannot identify the type of function, then there is nothing to do in Calculus.

Everything rests on identifying the function type, its category.

In Precalculus, we want to supply reasoning for our analysis choices. Identifying the category for the function often is the reaosning, because we have a list of characteristics for each function type.

We have three levels of function categories.

(a)

Elementary Function

(b)

Operation

(c)

Composition

1 Elementary Function

First Choice

If our function belongs to one of the elementary function categories, then that is important and very helpful.

CAN: This is a “can” question.

Our categories for Elementary Functions are

• Constant
• Linear
• Quadratic
• Polynomial
• Rational
• Radical or Root
• Exponential (Shifted)
• Logarithmic
• Absolute Value
• Power
• Sine
• Cosine
• Tangent

The standard forms for formulas look like

$$\begin {array}{ll} \text {Constant} & f(x) = A \\ \text {Linear} & f(x) = A x + B \\ \text {Quadratic} & f(x) = A x^2 + B x + C \\ \text {Polynomial} & f(x) = a_n x^n + \cdots + a_1 x + a_0 \\ \text {Rational} & f(x) = \frac {a_n x^n + \cdots + a_1 x + a_0}{b_m x^m + \cdots + b_1 x + b_0} \\ \text {Root} & f(x) = \sqrt [n]{A x + B} \\ \text {Exponential} & f(x) = A b^{B x + C} \\ \text {Shifted Exponential} & f(x) = A b^{B x + C} + D \\ \text {Logarithmic} & f(x) = A \log _b(B x + C) + D \\ \text {Absolute Value} & f(x) = A |B x + C| + D \\ \text {Power} & f(x) = x^A \\ \text {Sine} & f(x) = A \sin (B x + C) + D \\ \text {Cosine} & f(x) = A \cos (B x + C) + D \\ \text {Tangent} & f(x) = A \tan (B x + C) + D \end {array}$$

All of the insides are linear functions.

When trying to categorize a function, the formula you have may not be in one of these forms. That doesn’t matter. The question is whether or not, with a little algebra, you can get an equivalent formula that is in one of these forms.

This is our first choice. If we can identify our function as an elementary function, then that gives us the most information.

For us, in Precalculus, if the inside of the function is not linear, then it isn’t an elementary function.

2 Operation

Second Choice

IS: This is an “is” question.

$$f(x) + g(x)$$

$$f(x) - g(x)$$

$$f(x) \cdot g(x)$$

$$\frac {f(x)}{g(x)}$$

This is our second choice. If we can identify our function as an elementary function, then that gives us the most information. If not, then we would like some helpful information about its structure.

If we can view the function as an operation, then that is helpful information.

3 Composition

Third Choice

This is our default category. It signals to use the Chain Rule in Calculus.

When we view our function as a composition, then we would also like to know its component functions. These are usually elementary functions.

$$(f \circ g)(x) = f(g(x))$$

Through the component functions, we have a way to analyze compositions. We have a Precalculus version of the chain rule.

• $inc \circ inc = inc$
• $inc \circ dec = dec$
• $dec \circ inc = dec$
• $dec \circ dec = inc$

4 Piecewise Defined

Zeroth Choice

If a function is a piecewise defined function, then we usually see that right away, because the formula uses a big curly brace and a list of formulas and domain pieces.

$$f(x) = \begin {cases} formula 1 & domain 1 \\ formula 2 & domain 2 \\ formula 3 & domain 3 \\ formula 4 & domain 4 \end {cases}$$