{"id":563893,"date":"2021-12-22T11:30:27","date_gmt":"2021-12-22T11:30:27","guid":{"rendered":"https:\/\/www.indcareer.com\/schools\/?p=563893"},"modified":"2021-12-28T06:30:14","modified_gmt":"2021-12-28T06:30:14","slug":"maharashtra-board-solutions-class-11-arts-science-maths-part-2-chapter-6-functions","status":"publish","type":"post","link":"https:\/\/www.indcareer.com\/schools\/maharashtra-board-solutions-class-11-arts-science-maths-part-2-chapter-6-functions\/","title":{"rendered":"Maharashtra Board Solutions Class 11-Arts & Science Maths (Part 2): Chapter 6- Functions"},"content":{"rendered":"\n

Class 11: Maths Chapter 5 solutions. Complete Class 11 Maths Chapter 5 Notes.<\/p>\n\n\n\n

Maharashtra Board Solutions Class 11-Arts & Science Maths (Part 2): Chapter 6- Functions<\/h2>\n\n\n\n

Maharashtra Board 11th Maths Chapter 5, Class 11 Maths Chapter 5 solutions<\/p>\n\n\n\n

Ex 6.1<\/h4>\n\n\n\n

Question 1.
Check if the following relations are functions.
(a)<\/strong><\/p>\n\n\n\n

\"Maharashtra<\/figure>\n\n\n\n

Solution:
<\/strong>Yes.
Reason: Every element of set A has been assigned a unique element in set B.<\/p>\n\n\n\n

(b)<\/strong><\/p>\n\n\n\n

\"Maharashtra<\/figure>\n\n\n\n

Solution:
<\/strong>No.
Reason: An element of set A has been assigned more than one element from set B.<\/p>\n\n\n\n

(c)
<\/strong><\/p>\n\n\n\n

\"Maharashtra<\/figure>\n\n\n\n

Solution:
<\/strong>No.
Reason:
Not every element of set A has been assigned an image from set B.<\/p>\n\n\n\n

Question 2.
Which sets of ordered pairs represent functions from A = {1, 2, 3, 4} to B = {-1, 0, 1, 2, 3}? Justify.<\/strong>
(i) {(1, 0), (3, 3), (2, -1), (4, 1), (2, 2)}
(ii) {(1, 2), (2, -1), (3, 1), (4, 3)}
(iii) {(1, 3), (4, 1), (2, 2)}
(iv) {(1, 1), (2, 1), (3, 1), (4, 1)}
Solution:
<\/strong>(i) {(1, 0), (3, 3), (2, -1), (4, 1), (2, 2)} does not represent a function.
Reason:<\/strong> (2, -1), (2, 2), show that element 2 \u2208 A has been assigned two images -1 and 2 from set B.<\/p>\n\n\n\n

(ii) {(1, 2), (2, -1), (3, 1), (4, 3)} represents a function.
Reason: Every element of set A has been assigned a unique image in set B.<\/p>\n\n\n\n

(iii) {(1, 3), (4, 1), (2, 2)} does not represent a function.
Reason:
<\/strong>3 \u2208 A does not have an image in set B.<\/p>\n\n\n\n

(iv) {(1, 1), (2, 1), (3, 1), (4, 1)} represents a function
Reason: <\/strong>Every element of set A has been assigned a unique image in set B.<\/p>\n\n\n\n

Question 3.
Check if the relation given by the equation represents y as function of x.<\/strong>
(i) 2x + 3y = 12
(ii) x + y2<\/sup> = 9
(iii) x2<\/sup> \u2013 y = 25
(iv) 2y + 10 = 0
(v) 3x \u2013 6 = 21
Solution:<\/strong><\/p>\n\n\n\n

\"Maharashtra<\/figure>\n\n\n\n

(iii) x2<\/sup> \u2013 y = 25
\u2234 y = x2<\/sup> \u2013 25
\u2234 For every value of x, there is a unique value of y.
\u2234 y is a function of x.<\/p>\n\n\n\n

(iv) 2y + 10 = 0
\u2234 y = -5
\u2234 For every value of x, there is a unique value of y.
\u2234 y is a function of x.<\/p>\n\n\n\n

(v) 3x \u2013 6 = 21
\u2234 x = 9
\u2234 x = 9 represents a point on the X-axis.
There is no y involved in the equation.
So the given equation does not represent a function.<\/p>\n\n\n\n

\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n

Question 6.
Find x, if f(x) = g(x) where
(i) f(x) = x4<\/sup> + 2x2<\/sup>, g(x) = 11x2<\/sup>
(ii) f(x) = \u221ax \u2013 3, g(x) = 5 \u2013 x
Solution:<\/strong>
(i) f(x) = x4<\/sup> + 2x2<\/sup>, g(x) = 11x2<\/sup>
f(x) = g(x)
\u2234 x4<\/sup> + 2x2<\/sup> = 11x2<\/sup>
\u2234 x4<\/sup> \u2013 9x2<\/sup> = 0
\u2234 x2<\/sup> (x2<\/sup> \u2013 9) = 0
\u2234 x2<\/sup> = 0 or x2<\/sup> \u2013 9 = 0
\u2234 x = 0 or x2<\/sup> = 9
\u2234 x = 0, \u00b13<\/p>\n\n\n\n

(ii) f(x) = \u221ax \u2013 3, g(x) = 5 \u2013 x
f(x) = g(x)
\u2234 \u221ax \u2013 3 = 5 \u2013 x
\u2234 \u221ax = 5 \u2013 x + 3
\u2234 \u221ax = 8 \u2013 x
on squaring, we get<\/p>\n\n\n\n

\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n

f: N \u2192 N given by f(x) = x3<\/sup>
<\/p>\n\n\n\n

\"Maharashtra<\/figure>\n\n\n\n

\u2234 f is injective.
Numbers from codomain which are not cubes of natural numbers are not images under f.
\u2234 f is not surjective.<\/p>\n\n\n\n

(v) f : R \u2192 R given by f(x) = x3<\/sup>
Solution:<\/strong>
f: R \u2192 R given by f(x) = x3<\/sup>
<\/p>\n\n\n\n

\"Maharashtra<\/figure>\n\n\n\n

\u2234 For every y \u2208 R, there is some x \u2208 R.
\u2234 f is surjective.<\/p>\n\n\n\n

Question 14.
Show that if f : A \u2192 B and g : B \u2192 C are one-one, then gof is also one-one.
Solution:<\/strong>
f is a one-one function.
Let f(x1<\/sub>) = f(x2<\/sub>)
Then, x1<\/sub> = x2<\/sub> for all x1<\/sub>, x2<\/sub> \u2026..(i)
g is a one-one function.
Let g(y1<\/sub>) = g(y2<\/sub>)
Then, y1<\/sub> = y2<\/sub> for all y1<\/sub>, y2<\/sub> \u2026..(ii)
Let (gof) (x1<\/sub>) = (gof) (x2<\/sub>)
\u2234 g(f(x1<\/sub>)) = g(f(x2<\/sub>))
\u2234 g(y1<\/sub>) = g(y2<\/sub>),
where y1<\/sub> = f(x1<\/sub>), y2<\/sub> = f(x2<\/sub>) \u2208 B
\u2234 y1<\/sub> = y2<\/sub> \u2026..[From (ii)]
i.e., f(x1<\/sub>) = f(x2<\/sub>)
\u2234 x1<\/sub> = x2<\/sub> \u2026.[From (i)]
\u2234 gof is one-one.<\/p>\n\n\n\n

Question 15.
Show that if f : A \u2192 B and g : B \u2192 C are onto, then gof is also onto.
Solution:<\/strong>
Since g is surjective (onto),
there exists y \u2208 B for every z \u2208 C such that
g(y) = z \u2026\u2026.(i)
Since f is surjective,
there exists x \u2208 A for every y \u2208 B such that
f(x) = y \u2026\u2026.(ii)
(gof) x = g(f(x))
= g(y) \u2026\u2026[From (ii)]
= z \u2026..[From(i)]
i.e., for every z \u2208 C, there is x in A such that (gof) x = z
\u2234 gof is surjective (onto).<\/p>\n\n\n\n

\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n

\u2234 x < 2 or x > 3
Solution of (x \u2013 a) (x \u2013 b) > 0 is x < a or x > b where a < b
\u2234 Domain of f = (-\u221e, 2) \u222a (3, \u221e)<\/p>\n\n\n\n

\"Maharashtra<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n
\"Maharashtra<\/figure>\n\n\n\n

Download PDF<\/strong><\/h2>\n\n\n\n

Maharashtra Board Solutions Class 11-Arts & Science Maths (Part 2): Chapter 6- Functions<\/p>\n\n\n\n

Download PDF<\/strong>: Maharashtra Board Solutions Class 11-Arts & Science Maths (Part 2): Chapter 6- Functions PDF<\/a><\/p>\n\n\n\n

Chapterwise Maharashtra Board Solutions Class 11 Arts & Science Maths (Part 2) :<\/strong><\/h2>\n\n\n\n