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Additional resources for A history of the mathematical theory of probability : from the time of Pascal to that of Laplace
Identify any restrictions on the input. If there is a denominator in the function’s formula, set the denominator equal to zero and solve for x . If the function’s formula contains an even root, set the radicand greater than or equal to 0, and then solve. 3. Write the domain in interval form, making sure to exclude any restricted values from the domain. 18 Finding the Domain of a Function Involving a Denominator Find the domain of the function f (x) = x + 1 . 2−x Solution When there is a denominator, we want to include only values of the input that do not force the denominator to be zero.
Solve f (x) = 4. 38. Given the function f (x) = x + 2 : a. Evaluate f (7). 4 Chapter 1 Functions b. Solve f (x) = 4. 39. Consider the relationship 3r + 2t = 18. a. Write the relationship as a function r = f (t). b. Evaluate f ( − 3). c. Solve f (t) = 2. Graphical For the following exercises, use the vertical line test to determine which graphs show relations that are functions. 40. 41. 42. 39 40 43. 44. 45. 4 Chapter 1 Functions Chapter 1 Functions 46. 47. 48. 41 42 49. 50. 51. 4 Chapter 1 Functions Chapter 1 Functions 52.
Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function). Given the formula for a function, determine the domain and range. 1. Exclude from the domain any input values that result in division by zero. 2. Exclude from the domain any input values that have nonreal (or undefined) number outputs. 3. Use the valid input values to determine the range of the output values. 4. Look at the function graph and table values to confirm the actual function behavior.