Finding the Inverse of a Function
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Questions and Answers

If a function 𝑓 is invertible and (𝑎, 𝑏) is on its graph, what is the corresponding point on the graph of 𝑓 −1?

  • (−𝑏, 𝑎)
  • (𝑎, 𝑏)
  • (𝑏, 𝑎) (correct)
  • (𝑏, −𝑎)

    • What does the point (𝑏, 𝑎) represent on the graph of an inverse function if (𝑎, 𝑏) is a point on the graph of function 𝑓?

  • A point on the graph of 𝑓 −1 (correct)
  • A point on the graph of 𝑓
  • A non-existent point
  • A point outside the domain of 𝑓 −1

    • The relationship between the graphs of a function 𝑓 and its inverse 𝑓 −1 can be described as:

  • They are symmetrical about the x-axis.
  • They are identical graphs.
  • They are reflections over the y-axis.
  • They are symmetrical about the line 𝑦 = 𝑥. (correct)

    • If a function 𝑓 has an inverse, what can be inferred about the function's graph?

    <p>It has no horizontal segments.</p> Signup and view all the answers

    Given a point (𝑎, 𝑏) on the graph of an invertible function, what is the best method to find its equivalent point on the graph of 𝑓 −1?

    <p>Swap the coordinates to get (𝑏, 𝑎).</p> Signup and view all the answers

    What is the first step in finding the inverse of a function?

    <p>Let y equal f(x)</p> Signup and view all the answers

    When given the function g(x) = 2/(7x + 5), what do you obtain after interchanging variables in the first step?

    <p>x = 2/(7y + 5)</p> Signup and view all the answers

    What is the result of solving 7y + 5 = 2x for y in terms of x?

    <p>y = (2x - 5)/7</p> Signup and view all the answers

    In the function h(x) = 4 - 3x, what is the correct expression for h^{-1}(x) after finding its inverse?

    <p>(4 - x)/3</p> Signup and view all the answers

    After completing the steps to find the inverse of f(x) = 4050x, what is the expression for f^{-1}(x)?

    <p>x/4050</p> Signup and view all the answers

    What is achieved in the final step of finding the inverse function?

    <p>You replace y with f^{-1}(x).</p> Signup and view all the answers

    Which equation represents the proper interchange of x and y for the function given as y = 4 - 3x?

    <p>x = 4 - 3y</p> Signup and view all the answers

    In the function g(x) = 2/(7x + 5), what is the form of g^{-1}(x) after solving?

    <p>(2x - 5)/7</p> Signup and view all the answers

    How do you interpret f^{-1}(y) = x for a given function f?

    <p>f applies to x yielding y.</p> Signup and view all the answers

    Which step is involved when solving the equation 3y = 4 - x to isolate y?

    <p>Divide both sides by 3</p> Signup and view all the answers

    Study Notes

    Finding the Inverse of a Function

    • To find the inverse of a function, follow these steps:
      • Let y equal to the function f(x).
      • Interchange the variables x and y.
      • Solve for y in terms of x.
      • Replace y by f-1(x).
    • Note that f-1(y) = x, meaning the inverse function undoes what the original function does to a number within its domain.

    Examples of Finding the Inverse

    • Example 1: Find the inverse of f(x) = 4,050x.
      • Step 1: Let y = 4,050x
      • Step 2: Interchange x and y, so x = 4,050y
      • Step 3: Solve for y, resulting in y = x/4,050
      • Step 4: Replace y with f-1(x), giving f-1(x) = x/4,050
    • Example 2: Find the inverse of g(x) = (7x+5)/2.
      • Step 1: Let y = (7x + 5)/2
      • Step 2: Interchange x and y, resulting in x = (7y + 5)/2
      • Step 3: Solve for y, giving y = (2x - 5)/7
      • Step 4: Replace y with g-1(x), resulting in g-1(x) = (2x - 5)/7
    • Example 3: Find the inverse of h(x) = 4 - 3x
      • Step 1: Let y = 4 - 3x
      • Step 2: Interchange x and y, giving x = 4 - 3y
      • Step 3: Solve for y, resulting in y = (4 - x)/3
      • Step 4: Replace y with h-1(x), giving h-1(x) = (4 - x)/3

    Graphing Inverse Functions

    • If a function f is invertible and (a,b) is a point on the graph of f, then (b,a) is a point on the graph of f-1. This means the graph of f-1 is the reflection of the graph of f across the line y = x.
    • In other words, if you find a point (a, b) on f, you will find a point (b, a) on its inverse f-1.

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    Related Documents

    Inverse Functions PDF

    Description

    This quiz explores the process of finding the inverse of a function. You will learn how to interchange variables and solve for y in terms of x through a variety of examples. Test your understanding of function inverses and their properties.

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