Inverse Functions

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How to find inverse functions, including those with restricted domains

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Inverse Functions

  1. 1. Inverse Functions<br />Finding the Inverse<br />
  2. 2. 1st example, begin with your function <br /> f(x) = 3x – 7 replace f(x) with y<br /> y = 3x - 7<br />Interchange x and y to find the inverse<br /> x = 3y – 7 now solve for y<br /> x + 7 = 3y<br /> = y<br /> f-1(x) = replace y with f-1(x)<br />Finding the inverse<br />
  3. 3. 2nd example<br /> g(x) = 2x3 + 1 replace g(x) with y<br /> y = 2x3 + 1<br />Interchange x and y to find the inverse<br /> x = 2y3 + 1 now solve for y<br /> x - 1 = 2y3<br /> = y3<br />= y <br /> g-1(x) = replace y with g-1(x)<br />Finding the inverse<br />
  4. 4. Recall, to verify you have found the inverse you check that composition of the function with the inverse, in both orders, equals x<br />Using specific ordered pairs can illustrate how the inverse works, but does not verify that it is the inverse.<br />Verifying inverses<br />
  5. 5. Consider f(x) =<br />What is the domain?<br /> x + 4 &gt; 0<br /> x &gt; -4 or the interval [-4, ∞)<br />What is the range?<br /> y &gt; 0 or the interval [0, ∞)<br />Function with a restricted domain<br />
  6. 6. Now find the inverse:<br /> f(x) = D: [-4, ∞) R: [0, ∞)<br /> y =<br />Interchange x and y<br /> x = <br /> x2 = y + 4<br /> x2 – 4 = y<br /> f-1(x) = x2 – 4 D: [0, ∞) R: [-4, ∞)<br />Function with a restricted domain<br />
  7. 7. Finally, let us consider the graphs:<br /> f(x) =<br />D: [-4, ∞) R: [0, ∞)<br /> blue graph<br /> f-1(x) = x2 – 4 <br />D: [0, ∞) R: [-4, ∞)<br /> red graph<br />Functions with a restricted domain<br />
  8. 8. 2nd example<br />Consider g(x) = 5 - x2 D: [0, ∞)<br />What is the range? <br /> Make a very quick sketch of the graph<br /> R: (-∞, 5]<br />Function with a restricted domain<br />
  9. 9. Now find the inverse:<br /> g(x) = 5 - x2 D: [0, ∞) R: (-∞, 5]<br /> y = 5 - x2<br />Interchange x and y<br /> x = 5 - y2<br /> x – 5 = -y2<br /> 5 – x = y2<br /> = y<br />but do we want the + or – square root?<br /> g-1(x) = D: (-∞, 5] R: [0, ∞) <br />Function with a restricted domain<br />
  10. 10. And, now the graphs:<br /> g(x) = 5 - x2<br /> D: [0, ∞) R: (-∞, 5]<br /> blue graph<br /> g-1(x) = <br />D: (-∞, 5] R: [0, ∞) <br /> red graph<br />Functions with a restricted domain<br />
  11. 11. A function is one-to-one if each x and y-value is unique<br />Algebraically it means if f(a)=f(b), then a=b.<br />On a graph it means the graph passes the vertical and the horizontal line tests.<br />If a function is one-to-one it has an inverse function.<br />One-to-one<br />

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