Exponential functions
- General form for the exponential function: \(f\left(x\right)=Aa^x\)
- Most applications of the exponential function are used to take the time dimension into consideration: \(f\left(t\right)=Aa^t\)
- Interesting property: \(f\left(t+1\right)=Aa^{t+1}=Aa^ta\). Thus, \(f\left(t+1\right)=a\cdot f\left(t\right)\)\(a\)
Exponential model of population growth
- Using the same data from the example above, we can write: \(P\left(t\right)=641\cdot g^t\), where \(g\) is the population growth rate. What was the population growth rate for the ten years between 1960 and 1970 given that population in 1970 was 705 million?
The natural exponential function
- General formula of the natural exponential function: \(f\left(x\right)=e^x\)
- The power rules apply to the natural exponential function:
- \(e^se^t=e^{s+t}\)
- \(\frac{e^s}{e^t}=e^{s-t}\)
- \(\left(e^s\right)^t=e^{st}\)
Logarithmic functions
- We can use natural logs (and the rules associated with them) to solve exponential functions
- When \(e^x=a\) we say that \(x\) is the \(\ln\) of \(a\), or \(x=\ln a\).
Rules of log functions
- \(\ln\left(xy\right)=\ln x+\ln y\)
- \(\ln\frac{x}{y}=\ln\ x-\ln y\)
- \(\ln x^p=p\ln x\)
- \(\ln\ 1=0\)
- \(\ln e=1\)
- \(\ln e^x=x\)
Examples
- \(5e^{-3x}=16\)
- \(A\alpha e^{-\alpha x}=k\)
- \(e^x+4e^{-x}=4\)
Property of functions
Shifting graphs
- We will study the relationship between the following functions \(f\left(x\right),\ f\left(x\right)+c,\ f\left(x+c\right),\ cf\left(x\right),\ \) and \(f\left(-x\right)\)
- Take the following example: \(f\left(x\right)=\sqrt{x}\) and \(c=2\). The functions above become: \(f\left(x\right)+c=\sqrt{x}+2,\ f\left(x+c\right)=\sqrt{x+2},\ cf\left(x\right)=2\sqrt{x},\ f\left(-x\right)=\sqrt{-x}\)
- We can graph these using wolframalpha
Rules for shifting graphs of functions
- \(y=f\left(x\right)+c\): The graph for \(f\left(x\right)\) is moved up (down) by \(c\) if \(c>0\ \left(c<0\right)\)
- \(y=f(x+c)\): The graph is moved leftward (rightward) by \(c\) if \(c>0 (c<0)\)
- \(y=cf(x)\): The graph is stretched vertically if \(c>0\)
- \(y=f(-x)\): The graph is reflected about the \(y\) axis
Effect of a shift in supply
- At \(t=1\), \(D=a-bP\) & \(S=\alpha-\beta P\)
- Thus, the equilibrium price and and quantity are \(P^e=\frac{a-\alpha}{\beta-b}\) & \(Q^e=\frac{a\beta+\alpha b}{\beta+b}\)
- At \(t=2\), \(\bar{P}=\bar{a}-bP\)
- Solving for the new equilibrium condition yields: \(\bar{P}^e=\frac{a-\bar{\alpha}}{\beta+b}\) and \(\bar{Q}^e=\frac{a\beta+\bar{\alpha}b}{\beta+b}\)
- Find \(\bar{P}^e-P^e\) & \(\bar{Q}^e-Q^e\)
Tax credit vs. income deduction
- Let's consider the case of a progressive tax: \(T=f(y)\). Thus, the average tax rate in the economy is \(\frac{T}{y}=\frac{f(y)}{y}\)
- The government introduces the before-tax tax deduction \(d\). Thus, the taxable income becomes \(y-d\), giving us a tax function \(T=f(y-d)\)
- Alternatively, the government introduces the post-tax tax credit \(c\). The tax function becomes \(T=f(y)-c\)
- Numerical example: \(T=y^2, c=2, d=2\)
New functions from old
Composite functions
- If \(y\) is a function of \(u\) and \(u\) is a function of \(x\), we say that \(y\) is a composite function of \(x\):
- \(y=f(u)\) and \(u=g(x)\), \(y=f(g(x))\)
- \(g(x)\) is the interior function
- \(f(\cdot)\) is the exterior function
- Important rule: Whenever dealing with a composite function, first apply the rule identified by the interior function and then we apply the exterior function
Examples
- Write the following as composite functions
- \(y=(x^3+x^2)^{50}\)
- \(y=e^{-(x-\mu)^2}\)
Symmetry of function
- An even function is symmetric about the y-axis: \(f(x)=f(-x)\)
- An odd function is symmetric about the the origin: \(f(-x)=-f(x)\)
- A function can also be symmetric about a constant: \(f(a+u)=f(a-u)\)
Inverse functions
- Take the case of a demand function defined by: \(D=\frac{30}{P^{1/3}}\)
- We can express the same relationship but solving the equation above in terms of \(P\): \(P=\frac{2700}{D^3}\)
- We say that the latter formula is the inverse function of the former
- Important when dealing with inverse functions: the range of the two functions can (and often will be) different
Definition of inverse function
- Let \(f(\cdot)\) be a function with domain A and range B. If and only if \(f(\cdot)\) is one-to-one (if the function never gets the same value for two values of the domain), it has an inverse function \(g\left(\cdot\right)\) with domain B and range A
Examples
- Find the inverse functions for the following:
- \(y=4x-3\)
- \(y=\left(x+1\right)^{\frac{1}{5}}\)
- \(y=\frac{3x-1}{x+4}\)
Distance inn the plane: Circles
- The formula for the distance between two points on a plane is given by: \(d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}\)
- You can derive this from the Pythagorean theorem
- Example: \(P_1=\left(-4,\ 3\right),\ P_2=\left(5,\ -1\right)\)
- We can use this formula to derive the equation of a circle: Take a circle with radius \(r\) and center \(\left(a,\ b\right)\). The equation is then: \(r^2=\left(x-a\right)^2+\left(y-b\right)^2\)
General functions
- A function is a rule which to each element in a set A associates one and only one element in a set B.
- The general demand function for good x, for example, is really a function of, not only the price x, but also the income of the individual i, as well as the prices of all substitutes and complements of x: \(D_x^i:f\left(p_x,\ m_i,\ \vec{p}_{j\ne x}\right)\)
- If we denote a function by \(f\left(\cdot\right)\), the set A is called the domain of the function and the set B is called the target, where the rule expressed by the function assigns a unique each element of the domain to a unique element of the target. Thus, we call the target the image of the function