Math 101 Exam 1

Overview

Functions

• A function is a relation (or a rule) that assigns every input to only one output.
  • This means that if we could draw a vertical line on a graph that intersects the graph at more then one point, then there is an input value with more then one output, and hence the graph is not a function. (See tex'd version for picture.)

Functions over intervals

• Intervals are collections of input values.
  • The interval (-1,4) is all numbers $x$ so that $-1< x< 4$.
  • The interval [-2,$\infty$) is all numbers $x$ so that $x\geq -2$.
  • We always have the smaller numbers first in interval notation.
• If $f(x)$ is a function, $f(x)$ is increasing if the value (or height) of $f(x)$ increases as $x$ increases.
• If $f(x)$ is a function, $f(x)$ is decreasing if the value (or height) of $f(x)$ decreases as $x$ increases.
  • We can talk about what it means for $f(x)$ to be increasing or decreasing over an interval as well -- we just look at how the value changes for inputs in the interval.
• The average rate of change of a function $f(x)$ on the interval $[a,b]$ is $$\frac{f(b)-f(a)}{b-a}.$$

The value we get out the average rate of change ends up being the slope of the line between the points $(a,f(a))$ and $(b,f(b))$. (See tex'd version for image)

Linear Functions

• A linear function $f(x)$ is a function with a constant rate of change. That is, $\frac{f(b)-f(a)}{b-a}$ is the same for any $b$ and $a$. The graph of a linear function is always a line.
• Given a linear function $f(x)$,
  • The slope of $f(x)$ is given by $\frac{f(b)-f(a)}{b-a}$ for any $b$ and $a$.
  • The $x$-intercept is the input value where the line crosses the input axis. (And in general, for any function, no just linear ones, the $x$-intercepts are the input values where the function crosses the input axis.)
  • The $y$-intercept is the output value where the line crosses the output axis. (And in general, for any function, no just linear ones, the $y$-intercept is the output value where the function crosses the output axis.)
• We have two different forms for linear functions. They are each useful given specific information --
  • The slope-intercept form of a line is given by $f(x)=mx+b$, where $m$ is the slope and $b$ is the $y$-intercept.
    • When $m>0$, the line is increasing. When $m<0$, the line is decreasing. Otherwise, if $m=0$, the line is horizontal, with equation $f(x)=b$.
  • The point-slope form of a line is give by $y-y_0=m(x-x_0)$, where $m$ is the slope and $(x_0,y_0)$ is any point on the line.
• We can also talk about the relationship between two lines. Suppose $f(x)=m_1x+b_1$ and $g(x)=m_2x+b_2$ are two lines.
  • The lines are said to be parallel if $m_1=m_2$. If $b_1\neq b_2$, the lines never intersect. (Note that if $b_1=b_2$, and we have $m_1=m_2$, the lines are the same.)
  • The lines are said to be perpendicular if $m_1=-\frac{1}{m_2}$. The lines appear to form a 90 degree angle.

Domain and Range

• If $y=f(x)$ is a function, then
  • The domain of $f(x)$ is the collection of inputs $x$ that give an output value.
  • The range of $f(x)$ is the collections of outputs $y$ that correspond to the domain of $f(x)$. That is, the range is the collection of outputs of $f(x)$ when plugging in inputs from the domain.

Piecewise Functions

• The intervals next to the functions indicate for which input values one should use a specified function.
• See tex'd version for examples

Compositions and Inverses

• Omitted -- see exam 2 materials.

Exam 1 Topics and Sample Problems

Functions: e1rs2 2(a)

Average rate of change: e1rs1 2(a), e1rs2 1(c), e1rs2 7(b)

Parallel/Perpendicular: e1rs1 3, e1rs2 4

Linear Functions:

• Modeling: e1rs1 4(b), e1rs2 1(a), e1rs2 4(b)
• Interpreting: e1rs1 4(c)(d), 5(a)(b), e1rs2 4(c)(d)

Domain and Range: e1rs1 5(d), 6(a)(b)(d)(e)

Interpretation of functions: e1rs1 4(a), 6(c), e1rs2 2(b), e1rs2 4(a), e1rs2 7(a)

Compositions: e1rs1 7, e1rs2 8

Inverses: 2.2 problem 8

Piecewise Functions: e1rs1 6, e1rs2 5, e1rs2 6