Inverse of Exponential Functions Quiz

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Questions and Answers

What is the correct relationship between the graph of an exponential function and its inverse?

  • The graph of the inverse is a rotation of the original function by 90 degrees.
  • The graph of the inverse is identical to the graph of the original function.
  • The graph of the inverse is a reflection of the original function across the line y = x. (correct)
  • The graph of the inverse is a reflection of the original function across the line y = 2.

Which property must a function satisfy to have a unique inverse?

  • The function must have a range that includes every real number.
  • The function must be increasing over its entire domain.
  • The function must be defined for all real numbers.
  • The function must be one-to-one (injective). (correct)

What does the expression $y = ext{log}_b(x)$ represent?

  • The logarithm gives the exponent to which the base must be raised to obtain x. (correct)
  • The logarithm of x is equal to the base.
  • The value of y is the same as the value of x.
  • The base raised to the power of x equals y.

How can you solve an exponential equation like $2^x = 8$?

<p>By applying a logarithm with base 2 to both sides. (C)</p> Signup and view all the answers

What is the domain of the logarithmic function $y = ext{log}_b(x)$?

<p>x &gt; 0. (A)</p> Signup and view all the answers

What occurs when you compose a function and its inverse?

<p>The output is the original input value. (B)</p> Signup and view all the answers

Which of the following statements about logarithmic and exponential functions is true?

<p>Every exponential function has a corresponding logarithmic function as its inverse. (D)</p> Signup and view all the answers

If the base of a logarithmic function is negative, what can be concluded?

<p>The logarithmic function is not defined. (D)</p> Signup and view all the answers

Flashcards

Inverse of Exponential Function

A logarithmic function; if y = bˣ, then x = log₂(y).

Graph of Inverse Function

A reflection of the original function across the line y = x.

Solving Exponential Equations

Using logarithms to find the unknown variable in an exponential equation.

Logarithmic Function

The inverse of an exponential function; written as y = log₂(x).

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Properties of Inverse Functions

f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, and domain/range switch.

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One-to-one Function

A function where each output corresponds to only one input.

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Inverse Relation

A relationship that results when a function is not one-to-one.

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Common Logarithm

Logarithm with base 10 (log₁₀).

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Study Notes

Inverse of Exponential Functions

  • Exponential functions have a unique inverse, known as a logarithmic function.
  • The inverse relationship means that if y = bx, then x = logb(y), where b is the base.
  • The domain of an exponential function becomes the range of its inverse, and the range of the exponential function becomes the domain of its inverse.

Graphing Inverse Functions

  • The graph of an inverse function is a reflection of the original function across the line y = x.
  • To graph the inverse of a function, swap the x and y coordinates of points on the original graph.
  • The points on the inverse function are essentially the same as those on the original function, but mirrored across the line y = x.
  • A function and its inverse are symmetric about the line y = x.

Solving Exponential Equations

  • Exponential equations can often be solved using logarithms.
  • The key is to isolate the exponential term and then apply the logarithm with the same base as the exponential expression to both sides of the equation to solve for the unknown variable.
  • Applying the logarithm of a specific base (e.g., logarithms base 10 (common logarithm) or logarithms base e (natural logarithm)).
  • Techniques include applying the properties of logarithms and algebraic manipulation

Logarithmic Functions

  • Logarithmic functions are the inverses of exponential functions.
  • Logarithmic functions are defined as y = logb(x), where b is the base and x is the argument.
  • Logarithms represent the exponent to which the base must be raised to obtain a given number.
  • The logarithmic function y = logb(x) is defined only for x > 0 and b > 0, b ≠ 1.
  • Common logarithms are base-10 logarithms (log10). Natural logarithms are base-e logarithms (ln = loge).

Properties of Inverse Functions

  • The composition of a function and its inverse results in the identity function. f(f-1(x)) = x and f-1(f(x)) = x.
  • An inverse function, denoted by f-1(x), reverses the effect of the original function, f(x).
  • A function must be one-to-one (injective) to have an inverse function. This means each element in the range corresponds to exactly one element in the domain.
  • The domain of the inverse function is the range of the original function; and similarly the range of the inverse function is the domain of the original function.
  • If a function is not one-to-one, it does not have a unique inverse, rather it has an inverse relation.

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