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- Thread starter MattG03
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Let's first look at a graph of the two given functions and the given points:

From the graph, it appears the given points are where the two functions \(f\) and \(g\) intersect. Let's look at a table:

\(x\) | \(f(x)\) | \(g(x)\) |

-2 | 24 | 16 |

-1 | 10 | 10 |

0 | 0 | 6 |

1 | -6 | 4 |

2 | -8 | 4 |

3 | -6 | 6 |

4 | 0 | 10 |

5 | 10 | 16 |

6 | 24 | 24 |

7 | 42 | 34 |

The table would indicate that the two given points are where the two functions intersect. Now, we can verify this algebraically as follows:

\(\displaystyle f(x)=g(x)\)

\(\displaystyle 2x^2-8x=x^2-3x+6\)

Collect everything on the LHS:

\(\displaystyle x^2-5x-6=0\)

Factor:

\(\displaystyle (x+1)(x-6)=0\)

And so the values of \(x\) for which the two functions are equal are:

\(\displaystyle x\in\{-1,6\}\)

Let's verify by finding the values of the functions for those two values of \(x\):

\(\displaystyle f(-1)=2(-1)^2-8(-1)=2+8=10\)

\(\displaystyle g(-1)=(-1)^2-3(-1)+6=1+3+6=10\)

\(\displaystyle f(6)=2(6)^2-8(6)=72-48=24\)

\(\displaystyle g(6)=(6)^2-3(6)+6=36-18+6=24\)

And so we may conclude that the functions:

\(\displaystyle f(x)=2x^2-8x\) and \(\displaystyle g(x)=x^2-3x+6\)

intersect at the points:

\(\displaystyle (-1,10)\) and \(\displaystyle (6,24)\)

Does all that make sense?