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Finding the Inverse of an Exponential Or Logarithmic Function

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Finding the Inverse of an Exponential Or Logarithmic Function Lesson 32 Finding the Inverse of an Exponential or Logarithmic Function As I’ve mentioned a few different times while covering recent lessons, exponential functions and logarithmic functions are both one-to-one functions, so both have inverse functions. The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function. So when finding the inverse of an exponential function such 푓(푥) = 2푥, we simply convert that exponential function to a logarithmic function. 푓(푥) = 2푥 −1 푓 (푥) = log2(푥) Remember that the inverse of a function switches the inputs and outputs, so the domain of an exponential function is the same as the range of a logarithmic function, and the range of an exponential function is the same as the domain of a logarithmic function. 푓(푥) = 2푥 Domain of 푓: (−∞, ∞) Range of 푓: (0, ∞) (푥 can be any real number, but taking 2 to the power of 푥 only produces positive outputs) −1 −1 −1 푓 (푥) = log2(푥) Domain of 푓 : (0, ∞) Range of 푓 : (−∞, ∞) (푥 must be positive only, but the outputs can be any real values) Example 1: List the domain and range of the function 푓(푥) = log2(푥). Then find its inverse function 푓−1(푥) and list its domain and range. When finding the inverse −ퟏ of an exponential or 풇(풙) = 퐥퐨퐠ퟐ(풙) 풇 (풙) = logarithmic function, we are basically just converting from one form to the other. Later Domain of 풇: Domain of 풇−ퟏ: we’ll see examples where we need to add, subtract, multiply, and/or divide to move other terms and/or factors around, but with Range of 풇: Range of 풇−ퟏ: exponentials and logs we basically just convert from one form to the other. 1 Lesson 32 Finding the Inverse of an Exponential or Logarithmic Function Example 2: List the domain and range of the function 푓(푥) = log(푥) + 5. Then find its inverse function 푓−1(푥) and list its domain and range. a. 풇(풙) = 퐥퐨퐠(풙) + ퟓ Since this is a logarithmic function, the argument 푥 must be positive only (D: (0, ∞)) but the output log(푥) + 5 can be any real number (R: (−∞, ∞)). Domain of 풇: (ퟎ, ∞) Range of 풇: (−∞, ∞) To find the inverse, write the function as an equation and solve for 푥. a. 푓 = log10(푥) + 5 Remember that a logarithm is simply an exponent. log10(푥) represents the b. exponent that makes the base 10 equal to the argument 푥. Since 푓 − 5 is equal to log10(푥), it is the exponent that makes 10 equal to 푥, so that’s why 푓−5 푓 − 5 = log10(푥) we convert to 10 = 푥. Keep in mind that whenever you start with a logarithmic function, you will ALWAYS end up with an inverse that is an c. exponential function. The opposite is also true; anytime you have an 10푓−5 = 푥 exponential function, its inverse will ALWAYS be a logarithmic function. d. To undo + 5, we subtract 5; to undo log(푥), we convert to exponential form e. 풇−ퟏ(풙) = ퟏퟎ풙−ퟓ Since this is an exponential function, the input 푥 can be any real number (D: (−∞, ∞)) and the output 10푥−5 is a power of base 10, so it is positive only (R: (0, ∞)). Domain of 풇−ퟏ: (−∞, ∞) Range of 풇−ퟏ: (ퟎ, ∞) When finding the inverse of an exponential or logarithmic function, we are basically just converting from one form to the other. If you start with logarithmic function, its inverse will ALWAYS be exponential, and if you start with an exponential function, you will ALWAYS end up with a log function. 2 Lesson 32 Finding the Inverse of an Exponential or Logarithmic Function Example 3: List the domain and range of the function 푓(푥) = ln(−푥). Then find its inverse function 푓−1(푥) and list its domain and range. a. 풇(풙) = 퐥퐧(−풙) Since this is a logarithmic function, the argument −푥 must be positive only (−푥 > 0). If −푥 > 0, then 푥 < 0, so the domain in the case is negative values only (D: (0, ∞)). The output ln(−푥) can be any real number (R: (−∞, ∞)). Domain of 풇: (−∞, ퟎ) Range of 풇: (−∞, ∞) To find the inverse, write the function as an equation and solve for 푥. f. 푓 = ln(−푥) g. 푓 = log푒(−푥) A logarithm is an exponent. Anything that is equal to a logarithm is also an exponent. So if you have an equation such as 푓 = log푒(−푥), that means h. that 푓 is an exponent. More specifically, 푓 is the exponent that makes 푒 푒푓 = −푥 equal to −푥. −푒푓 = 푥 i. To undo ln(−푥), we convert to exponential form; to undo −푥, we negate both sides of the equation j. 풇−ퟏ(풙) = −풆풙 Since this is an exponential function, the input 푥 can be any real number (D: (−∞, ∞)). The output 푒푥 is a power of base 푒, so it produces positive outputs only. However negating it to get −푒푥 means we end up with negative outputs only (R: (− ∞, 0)). Domain of 풇−ퟏ: (−∞, ∞) Range of 풇−ퟏ: (−∞, ퟎ) 3 Lesson 32 Finding the Inverse of an Exponential or Logarithmic Function Example 4: List the domain and range of the function 푓(푥) = 2 ∙ log(푥 − 3) + 4. Then find its inverse function 푓−1(푥) and list its domain and range. k. 풇(풙) = ퟐ ∙ 퐥퐨퐠(풙 − ퟑ) + ퟒ 풇−ퟏ(풙) = Domain of 풇: Range of 풇: Domain of 풇−ퟏ: Range of 풇−ퟏ: 4 Lesson 32 Finding the Inverse of an Exponential or Logarithmic Function Example 5: List the domain and range of the function 1 푓(푥) = − ∙ log (1 − 푥). Then find its inverse function 푓−1(푥) and list 2 2 its domain and range. l. ퟏ 풇(풙) = − ∙ 퐥퐨퐠 (ퟏ − 풙) 풇−ퟏ(풙) = ퟐ ퟐ Domain of 풇: Range of 풇: Domain of 풇−ퟏ: Range of 풇−ퟏ: 5 Lesson 32 Finding the Inverse of an Exponential or Logarithmic Function 1 푥 Example 6: List the domain and range of the function 푓(푥) = ( ) . Then 2 find its inverse function 푓−1(푥) and list its domain and range. b. v ퟏ 풙 풇(풙) = ( ) ퟐ Since this is an exponential function, the input 푥 can be any real number (D: (−∞, ∞)) and the output 2푥 is a power of base 2, so it is positive only (R: (0, ∞)). Domain of 풇: (−∞, ∞) Range of 풇: (ퟎ, ∞) To find the inverse, write the function as an equation and solve for 푥. 1 푥 푓 = ( ) 2 a. 1 푥 To undo ( ) we convert to logarithmic form 2 log1(푓) = 푥 2 −ퟏ 풇 (풙) = 퐥퐨퐠ퟏ(풙) ퟐ Since this is a logarithmic function, the argument 푥 must be positive only (D: (0, ∞)) but the output log1(푥) can be any real number (R: (−∞, ∞)). 2 Domain of 풇−ퟏ: (ퟎ, ∞) Range of 풇−ퟏ: (−∞, ∞) Again, when finding the inverse of an exponential or logarithmic function, we are basically just converting from one form to the other. 6 Lesson 32 Finding the Inverse of an Exponential or Logarithmic Function Example 7: List the domain and range of the function 푓(푥) = 푒푥+3. Then find its inverse function 푓−1(푥) and list its domain and range. c. v 풇(풙) = 풆풙+ퟑ Since this is an exponential function, the input 푥 can be any real number (D: (−∞, ∞)) and the output 푒푥+3 is a power of base 푒, so it is positive only (R: (0, ∞)). Domain of 풇: (−∞, ∞) Range of 풇: (ퟎ, ∞) To find the inverse, write the function as an equation and solve for 푥. 푓 = 푒푥+3 a. log푒(푓) = 푥 + 3 b. log푒(푓) − 3 = 푥 c. To undo 푒 to a power, we convert to log form; to undo +3, we subtract 3 d. 풇−ퟏ(풙) = 퐥퐧(풙) − ퟑ Since this is a logarithmic function, the argument 푥 must be positive only (D: (0, ∞)) but the output ln(푥) − 3 can be any real number (R: (−∞, ∞)). Domain of 풇−ퟏ: (ퟎ, ∞) Range of 풇−ퟏ: (−∞, ∞) 7 Lesson 32 Finding the Inverse of an Exponential or Logarithmic Function Example 8: List the domain and range of the function 푓(푥) = 10푥 − 5. Then find its inverse function 푓−1(푥) and list its domain and range. a. 풇(풙) = ퟏퟎ풙 − ퟓ Since this is an exponential function, the input 푥 can be any real number (D: (−∞, ∞)). The output 10푥 − 5 can be broken into two parts, 10푥 is a power of base 10, so it is always positive. Taking 10푥 and subtracting 5 from it means taking our positive outputs and subtracting 5 from them, so we end up with outputs going from −5 to infinity (R: (−5, ∞)). Domain of 풇: (−∞, ∞) Range of 풇: (−ퟓ, ∞) To find the inverse, write the function as an equation and solve for 푥. a. 푓 = 10푥 − 5 b. 푓 + 5 = 10푥 c. log10(푓 + 5) = 푥 d. 풇−ퟏ(풙) = 퐥퐨퐠(풙 + ퟓ) Since this is a logarithmic function, the argument 푥 + 5 must be positive only, so we have 푥 + 5 > 0, which is 푥 > −5 (D: (−5, ∞)). The output log(푥 + 5) can be any real number (R: (−∞, ∞)). Domain of 풇−ퟏ: (−ퟓ, ∞) Range of 풇−ퟏ: (−∞, ∞) 8 Lesson 32 Finding the Inverse of an Exponential or Logarithmic Function Example 9: List the domain and range of the function 푓(푥) = 7 − 23푥+5. Then find its inverse function 푓−1(푥) and list its domain and range.
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