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## Algebra: Functions Test-2

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Question 1 |

*f*(x) = 2x +3 and

*g*(x) = (x -3)/2 thenWhat is the value of

*fo (fog) o (gof) (x)*

x | |

x ^{2} | |

2x+3 | |

$ \displaystyle \frac{x+3}{4x-5}$ |

Question 1 Explanation:

*fog(x)=gof (x)=x*

*fo (fog) o (gof) (x) = fo (fog) (x) =f (x)=2x+3.*

Therefore (c) is the correct answer.

Question 2 |

If le (x, y) = least of (x, y), mo (x) = I X I,me (x, y) = maximum of (x, y).

Find the value of me

Find the value of me

*(a+ mo (Ie (a, b)); mo (a+ me (mo (a)mo (b))),*at a = - 2 and b =-31 | |

0 | |

5 | |

3 |

Question 2 Explanation:

Segregating and Simplifying:

i)mo (a+ me (mo (a)mo (b)) =mo (a+ me (mo (a),3)

=mo (a+ me (2,3))

=mo (a+ 3)

= |-2+3|

=1

ii) a+ mo (Ie (a, b))

=a+mo(-3)

=-2+3

=1

i)mo (a+ me (mo (a)mo (b)) =mo (a+ me (mo (a),3)

=mo (a+ me (2,3))

=mo (a+ 3)

= |-2+3|

=1

ii) a+ mo (Ie (a, b))

=a+mo(-3)

=-2+3

=1

*The maximum value of both the terms is 1.*

Question 3 |

If le (x, y) = least of (x, y), mo (x) = I X I,me (x, y) = maximum of (x, y)
Which of the following must always be correct for a, b >0

mo (le (a, b)) > (me (mo (a),mo (b))) | |

mo (le (a, b))> (me (mo (a),mo (b)) | |

mo (le (a, b))< (Ie(mo (a)), mo (b)) | |

mo (le (a, b)) = le (mo (a),mo (b)) |

Question 3 Explanation:

First observation since both a and b are greater than 0,

mo(a)=a and mo(b)=b.

Therefore,

(me (mo (a),mo (b))) = me (a,b)

le (mo (a),mo (b))=le (a,b)

Now since both a and b are positive,

we can conclude that mo (le (a, b)) = le (a, b)

mo(a)=a and mo(b)=b.

Therefore,

(me (mo (a),mo (b))) = me (a,b)

le (mo (a),mo (b))=le (a,b)

Now since both a and b are positive,

we can conclude that mo (le (a, b)) = le (a, b)

Question 4 |

If le (x, y) = least of (x, y), mo (x) = I X I,me (x, y) = maximum of (x, y)
For what values of a is me (a

^{2}- 3a, a - 3) <0? a<0 anda<3 | |

a<0 ora<3 |

Question 4 Explanation:

**Case I**. a < 0, a

^{3}- 3a > a - 3 = a (a - 3) < 0 or 0 < a < 3 which is not true.

**Case II**. 0 < a < 3, a (a - 3) < 0 or 0 < a < 3 which is true.

**Case III**. a = 3, me (0, 0) < 0 not true.

**Case IV**. a> 3, a (a - 3) < 0 or 0 < a < 3 not true

The above graph clearly shows that the maximum value of the function occurs when a lies between 0 and 3. Therefore, option b is correct. The above graph clearly shows that the maximum value of the function occurs when a lies between 0 and 3.

Question 5 |

If le (x, y) = least of (x, y), mo (x) = I X I,me (x, y) = maximum of (x, y)
For what values of a le (a

^{2}- 3a, a - 3) < 0a | |

a |

Question 5 Explanation:

Again in case I, a < 0; a - 3 < 0 or a < 3

(from last Question) can be true

In case II, 0 < a < 3; a - 3 < 0 or a < 3 can be true

In case III, a = 3, le (0, 0) = 0 < 0, not true

In case IV, a> 3, a - 3 < 0 or a < 3 not true

The graph clearly shows that the minimum value of the function lies between 0 and 3. Therefore the answer will be option (b).

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