• We have $f(-2) = (-2)^3 = -8$ because $x=-2$ satisfies $x<-1$.
• Note that $-1$ does not lie in any of the three given regions, which means $f(-1)$ is undefinded.
• We know $f(4) = \sqrt{4} = 2$ as $x=4$ satisfies $x\geq 4$.

Given $f(x) = x^2$, we know $f(3) = 9$, $f(4) = 16$, and $f(3+4)=f(7)=49$. Since $9+16 = 25\neq 49$, then we know $f(3+4) \neq f(3) + f(4)$.

Important take away:

• In general, the square of a sum is NOT the sum of the squares, $(a + b)^2 \neq a^2 + b^2$.
• The operation of addition must happen before the power. The order of operations is grouping and then exponents.

There is an entire class of functions for which $f(a+b) = f(a)+f(b)$. We call these "linear functions" and they are studied in depth in Math 341: Linear Algebra.

A line is a function that can be written in the form $f(x) = mx+b$, where $m$ and $b$ are constants.

Note: The input to this function $x$ is a variable and the constants $m$ and $b$ are parameters of this function $f$. To make this explicit mathematically we can write $f(x; m, b) = mx + b$. The colon separates the variables from the parameters.

A quadratic function is a function that can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants with $a \neq 0$.

A polynomial function is any function that can be written in the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = \sum_{k=0}^n a_kx^k$ for some integer $n \geq 0$ and constants $a_0$, $a_1$, ... $a_{n-1}$, $a_n$ (with $a_n \neq 0$). The integer $n$ we call the degree of the polynomial.

When $n=0$, we relax the condition ($a_n \neq 0$) and allow $a_0 = 0$. The function $f(x) = 0$ is called the zero function.

A power function is any function of the form $f(x) = ax^b$ where $a\neq 0$ and $b$ are real numbers.

A root function is a power function of the form $f(x) = x^{1/n}$ where $n$ is a positive integer greater than one.

A rational function is any function of the form $f(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials.

An exponential function is any function of the form $f(x) = b^x$ where $b > 0$ and $b \neq 1$.

The function $w(x) = 5^x$ is an exponential function.

These are some solutions to the questions above. Come up with your own, and jot them on the chalk board.

• Examples of polynomials that are NOT power functions.
  • $q(x) = x^3 - 3x + 2$
  • $A(x) = (x-1)^2$
• Examples of power functions that are NOT polynomials
  • $g(x) = \sqrt{x}$
  • $f(x) = -2x^{-4}$

A function $f$ consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output. The set of inputs is called the domain of the function. The set of outputs is called the range of the function.

A function from set D to a set Y is a rule that assigns each element in set D exactly one element from set Y.

Note:

• The domain of a function from set D to a set Y is the set of inputs of the function. Notice all the values of set D must be used as inputs so the domain is the set D.
• The range of a function from set D to a set Y is the set of outputs of the function. Notice the range of the function is not necessarily set Y, not all the value of set Y must be used as outputs.

A function is a rule that assigns to every value of the domain just one value in the range. The domain is the inputs of the function. The range is the outputs of the function.