EPPS Math and Coding Camp
Functions and Relations - Aug 8
3.1 FUNCTIONS
- A relation that assigns one element of the range to each element of the domain is a function.
- One that assigns a subset of the range to each element of the domain is a correspondence.
3.1.1 Equations
- implicit notation \[ y=f(x) \]
- explicit notation
Example: \(y=3x\)
\(y\) is a function of \(x\);
\(x\) is the argument of the function
3.1.2 Graphs
3.1.3 Some Properties of Functions
- We define the function \(f\) as \(f(x) : A \rightarrow B\). This is often read as “\(f\) maps \(A\) into \(B\).” \(A\) here is the domain of the function, \(B\) is known as the codomain.
- The set of all values actually reached by running each \(x \in A\) through \(f\) is known as the image, or range, and it is necessarily a subset of B.
- function composition: \(g \circ f(x)\) or \(g(f(x))\)
3.1.3.1 Identity and Inverse Functions
| Term | Meaning |
|---|---|
| Identity function | Elements in domain are mapped to identical elements in codomain |
| Inverse function | Function that when composed with original function returns identity function |
| Surjective (onto) | Every value in codomain produced by value in domain |
| Injective (one-to-one) | Each value in range comes from only one value in domain |
| Bijective (invertible) | Both surjective and injective; function has an inverse |
3.1.3.2 Monotonic Functions
| Term | Meaning |
|---|---|
| Increasing | Function increases on subset of domain |
| Decreasing | Function decreases on subset of domain |
| Strictly increasing | Function always increases on subset of domain |
| Strictly decreasing | Function always decreases on subset of domain |
| Weakly increasing | Function does not decrease on subset of domain |
| Weakly decreasing | Function does not increase on subset of domain |
| (Strict) monotonicity | Order preservation; function (strictly) increasing over domain |
3.2.1 The Linear Equation (Affine Function)
- \(y=a+bx\)
- A linear equation is an equation that contains only terms of order \(x^1\) and \(x^0=1\). In other words, only \(x\) and \(1\), multiplied by constants, may appear on the right-hand side (RHS) of a linear equation.
3.2.2 Linear Functions
- Additivity: \(f(x_1+x_2)=f(x_1)+f(x_2)\)
- Scaling (aka homogeneity):\(f(ax)=af(x), \text{for all a}\).
- Linear functions are not affine functions, e.g., they do not permit a translation(the \(x^0\) term).
3.2.3 Nonlinear Functions:
3.2.3.2 Quadratic Functions
\[ y=\alpha +\beta_1x+\beta_2x^2 \]
3.2.3.4 Logarithms (Properties of log)
- Logarithms: \(log_a x\)
- \(a^{log_a x}=x\)
- \(log_a a^x=x\)
- if \(log_a x=b\), then \(a^{log_ax}=a^b\), thus \(x=a^b\)
- The base for the natural log is the \(e \approx 2.7183\)
Example:
If you suspect that education increases the probability of voting in national elections, but that each additional year of education has a smaller impact on the probability of voting than the preceding year’s, then the log functions are good candidates to represent that conjecture.
if \(p_v\) is “probability of voting” and \(ed\) is “years of education”:
- then \(p_v= \alpha + \beta ed\) specifies a linear relationship where an additional year of education has the same impact on the probability of voting regardless of how many years of education one has had.
- \(p_v= \alpha + \beta ed^2\) represents the claim that the impact of education on the probability of voting rises the more educated one becomes.
- We transform the relationship between \(p_v\) and \(ed\) from a linear one to a nonlinear one where the impact of an additional year of education declines the more educated one becomes: \(p_v= \alpha + \beta (\ln(ed))\)
- Algebraic rules for logs
\[ \ln(x_1 \cdot x_2)=\ln(x_1)+\ln(x_2) \text{, for }x_1,x_2>0 \]
\[ \ln \frac{x_1}{x_2}=\ln(x_1)-\ln(x_2) \text{, for }x_1,x_2>0 \]
\[ \ln(x_1 + x_2) \neq \ln(x_1)+\ln(l_2) \text{, for }x_1,x_2>0 \]
\[ \ln(x_1 - x_2) \neq \ln(x_1)-\ln(lx) \text{, for }x_1,x_2>0 \]
\[ \ln(x^b)=b\ln(x) \text{, for } x>0 \]
\[ \ln(1+x) \approx x \text{, for } x>0 \text{ and } x \approx 0 \]
3.3.1 Preference Relations
Transitivity states that if \(a\) is at least as good as \(b\), and \(b\) is at least as good as \(c\), then a is at least as good as \(c\): if \(aRb\) and \(bRc\), then \(aRc\).
Symmetry states that if \(aRb\), then \(bRa\) for all \(a\) and \(b\). In the realm of preference orderings, this implies complete indifference: everything is at least as good as everything else.
Reflexivity: A relation on a set \(A\) is reflexive if for all \(a \in A\), \(aRa\) is \(True\).
3.3.2 Utility Functions
- Integers and the real numbers are complete and transitive for all the usual ordering relations.
- Thus, if we want to represent our “at least as good as” relation with numbers, this relation had better have the same properties.
- we can translate the relation R on any set A to a function u on the same set.
- This u is called a utility function and assigns a value, typically a real number, to each element in A.
Problems
Problem 1
If the domain of the function \(y = 5 + 3x\) is the set \(\{x \mid 1 < x < 4\}\), find the range of the function and express it as a set.
Problem 2
For the function \(y = -x^2\), if the domain is the set of all nonnegative real numbers, what will its range be?
Problem 3
Graph the functions:
\(y = -x^2 + 5x - 2\)
\(y = x^2 + 5x - 2\)
with the set of values \(-5 < x < 5\) as the domain. It is well known that the sign of the coefficient of the \(x^2\) term determines whether the graph of a quadratic function will have a “hill” or a “valley.” On the basis of the present problem, which sign is associated with the hill? Supply an intuitive explanation for this.
Problem 4
Graph the function \(y = \frac{36}{x}\), assuming that \(x\) and \(y\) can take positive values only. Next, suppose that both variables can take negative values as well; how must the graph be modified to reflect this change in assumption?
Problem 5
Simplify \(\frac{x^3}{x-3}\).
Problem 6
Show that \(x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m\).
Problem 7
Simplify \(h(x) = g(f(x))\), where \(f(x) = x^2 + 2\) and \(g(x) = px - 4\).
Problem 8
Simplify \(h(x) = f(g(x))\) with the same \(f\) and \(g\). Is it the same as your previous answer?
Any Questions?
