Worksheet: Inverses of Exponential Functions

In this worksheet, we will practice dealing with the inverse of the exponential function, which is called the logarithm function.

Q1:

The function 𝑓 ( 𝑥 ) = 2 𝑒 + 3 𝑥 has an inverse of the form 𝑔 ( 𝑥 ) = ( 𝑎 𝑥 + 𝑏 ) l n . What are the values of 𝑎 and 𝑏 ?

  • A 𝑎 = 3 2 , 𝑏 = 1 2
  • B 𝑎 = 1 , 𝑏 = 3
  • C 𝑎 = 2 , 𝑏 = 3
  • D 𝑎 = 1 2 , 𝑏 = 3 2
  • E 𝑎 = 1 2 , 𝑏 = 3 2

Q2:

Make 𝑥 the subject of the equation 𝑦 = 2 𝑎 𝑥 + 𝑏 , assuming that 𝑎 0 .

  • A 𝑥 = ( 𝑦 ) + 𝑏 𝑎 l o g 2
  • B 𝑥 = ( 𝑦 ) 𝑏 l o g 2
  • C 𝑥 = ( 𝑦 2 ) 𝑏 𝑎 l o g
  • D 𝑥 = ( 𝑦 ) 𝑏 𝑎 l o g 2
  • E 𝑦 = ( 𝑥 ) 𝑏 𝑎 l o g 2

Q3:

Rearrange the equation 𝑦 = 2 1 2 + 1 𝑥 + 1 𝑥 to find 𝑥 in terms of 𝑦 . Hence determine the inverse 𝑔 1 to the function 𝑔 ( 𝑥 ) = 2 1 2 + 1 𝑥 + 1 𝑥 .

  • A 𝑔 ( 𝑥 ) = 𝑥 + 1 𝑥 2 1 2 l o g
  • B 𝑔 ( 𝑥 ) = 𝑥 1 𝑥 2 1 l o g
  • C 𝑔 ( 𝑦 ) = 𝑦 1 𝑦 2 1 l o g
  • D 𝑔 ( 𝑥 ) = 𝑥 1 𝑥 2 1 2 l o g
  • E 𝑔 ( 𝑦 ) = 𝑦 1 𝑦 2 1 2 l o g

Q4:

Consider the function 𝑓 ( 𝑥 ) = 𝑏 , where 𝑏 is a positive real number not equal to 1. What is the domain of 𝑓 ( 𝑥 ) ?

  • A 0 < 𝑥 < 𝑏
  • B 𝑥 > 𝑏
  • C all real numbers
  • D 𝑥 > 0

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