Let \(f(x)\) be an odd function defined for all real numbers \(x,\) and let \(g(x) = f(x + 3) - 5.\) You are told that the graph of \(y = g(x)\) passes through the point \((2,-2).\) Then the graph of \(y = g(x)\) must also pass through two other points \((a,b)\) and \((c,d).\) Find \((a,b)\) and \((c,d).\)
+9466 | Since f(x) is an odd function defined for all real numbers x ,
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| f( -x ) = - f(x) | ___ | for all real numbers x . And 0 is a real number, so.... |
| f( -0 ) = - f(0) |
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| f(0) = - f(0) |
| Here we can notice that the only way for a = -a to be true is if a = 0 . Still we can add f(0) to both sides of the equation. |
| 2f(0) = 0 |
| Divide both sides of the equation by 2 |
| f(0) = 0 |
| Now let's make this match the form f(x + 3) - 5 . Rewrite 0 as -3 + 3 |
| f( -3 + 3 ) = 0 |
| Subtract 5 from both sides of the equation. |
| f( -3 + 3 ) - 5 = -5 |
| Substitute g( - 3 ) in for f(-3 + 3) - 5 |
| g( -3 ) = -5 |
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| Now we can see that y = g( x ) passes through the point (-3, -5) . This makes sense because the graph of g(x) is shifted 3 to the left and 5 down from f(x) .
Since y = g(x) passes through (2, -2) , | ||
| g( 2 ) = -2 |
| Substitute f(2 + 3) - 5 in for g( 2 ) |
| f( 2 + 3 ) - 5 = -2 |
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| f( 5 ) - 5 = -2 |
| Add 5 to both sides. |
| f( 5 ) = 3 |
| Notice that (5, 3) is shifted 3 to the right and 5 up from (2, -2). Since f( -x ) = - f( x ) , f( -5 ) = - f(5) = -3 |
| f( -5 ) = - 3 |
| Rewrite -5 as -8 + 3 |
| f( -8 + 3 ) = -3 |
| Subtract 5 from both sides of the equation. |
| f( -8 + 3 ) - 5 = -8 |
| Substitute g( -8 ) in for f( -8 + 3 ) - 5 |
| g( -8 ) = -8 |
| And notice (-8, -8) is shifted 3 to the left and 5 down from (-5, -3) . |
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Now we can see that y = g( x ) passes through the point (-8, -8) . | ||
