- The total cost of producing \(x\) units of a product is given by \(C\left(x\right)=100x\sqrt{x}+500\). What happens if \(x=16\)? What is \(x=a+1\)?
- Suppose now that the cost function for the same product is given by \(C\left(x\right)=Ax\sqrt{x}+b\). What is the total cost of producing 0, 10, and \(x+h\) units?
Domain and range
- The domain of a function must always be specified
- As a general rule, "if a function is defined using an algebraic formula, the domain consists of all values of the independent variable for which the formula gives a unique value (unless another domain is explicitly mentioned)."
Examples
- Find the domain for \(f\left(x\right)=\frac{1}{x+3}\)
- Find the domain for \(g\left(x\right)=\sqrt{2x+4}\)
- Show that \(4\) is in the range of \(g\left(x\right)\)
Increasing and decreasing functions
- We say that a function is increasing if \(x_1\leq x_2\) implies \(f\left(x_1\right)\le f\left(x_2\right)\) \(\forall x_1,\ x_2\)
- We say that a function is strictly increasing if \(x_1<x_2\) implies \(f\left(x_1\right)<f\left(x_2\right)\) \(\forall x_1,\ x_2\)
Graphs of functions
- The Cartesian coordinate system