# Functions, Limits, and Continuity

## Contents

#### Problems

Problem : Could the following be the graph of a function f (x) ?

Figure %: Is this graph a function f (x) ?

No. For each positive x value, the graph contains two values for f (x) (namely, ± . Since a function takes on only one value for each x in its domain, this cannot be the graph of a function.

Problem : Find a linear function that passes through the points (1, 3) and (- 2, - 3)

We substitute (x 1, y 1) = (1, 3) and (x 2, y 2) = (- 2, - 3) into the @@equation@@ given in this section to obtain

f (x) = (x - 1) + 3 = 2x + 1

Problem : Use a power function to solve the following problem. A mathematician named George mows lawns all summer and manages to save up 3, 000 dollars. George decides to invest his earnings in an account that pays an annual interest rate of 7 percent. How much money will George have in his account after 5 years (assuming he does not make any further deposits)?

We may use a power function to describe this situation because we want to study a quantity that is being multiplied by a fixed number each year. The initial value in this problem is 3000 (in dollars), and the growth rate is 1.07 (per year). Thus the appropriate power function is f (t) = 3000(1.07)t , where t is the number of years from the time the money is invested. Plugging in 5 for t , we see that after 5 years George will have f (5) = 3000(1.07)5 4207.66 dollars.