117227
If \(f: A \rightarrow B\) and \(g: B \rightarrow C\) are two functions such that gof : \(A \rightarrow C\) is a bijections, then which one of the following is always true?
1 f and g are bijections
2 \(f\) is an injection and \(g\) is a surjection
3 \(f\) is a surjection and \(g\) is an injection
4 \(f\) is a bijection but \(g\) is not a bijection
Explanation:
B \(\operatorname{gof}(\mathrm{y})=\mathrm{z}\) \(g(f(y))=z\) \(f\) lies range but \(\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}\) for \(a, y \in A, f(y) \in B\), let \(f(y)=x\) For every element \(\mathrm{z} \in \mathrm{c}\), we have at \(\mathrm{x} \in \mathrm{B}\). Such that \(\mathrm{g}(\mathrm{x})=\mathrm{z}\) \(\therefore \quad \mathrm{g}\) is onto. Hence, \(\mathrm{f}\) is an injection and \(\mathrm{g}\) is a surjection.
Shift-I
Sets, Relation and Function
117229
Which of the following is true?
1 The composition of function is commutative
2 Every function is invertible
3 If a function \(f\) is bijective then its inverse \(\mathrm{f}^{-1}\) need not be bijective
4 The composition of functions is associative
Explanation:
D Consider \(\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}\) and \(\mathrm{g}: \mathrm{B} \rightarrow \mathrm{C}\) be two functions. Then the composition of \(f\) and \(g\) denoted by gof. is defined as the function gof: \(\mathrm{A} \rightarrow \mathrm{C}\) given by go gof \((x)=g f(x)\) Properties of function compositions. Associative property : Associative property of function composition, If there are three functions \(\mathrm{f}, \mathrm{g}\) and \(\mathrm{h}\), then they are said to be associative if and only is - \(\text { fo }(g o h)=(f o g) \text { oh. }\) The function composition of two onto function is always onto. The function composition of one - to - one function is always onto. The function composition of one- to-one function is always one to one So, the composition of function is associative is true statement.
JCECE-2018
Sets, Relation and Function
117235
If \(f(x)=\frac{4 x+7}{7 x-4}\), then the value of \(\mathbf{f}\{\mathbf{f}[\mathbf{f}(\mathbf{2})]\}=\)
117227
If \(f: A \rightarrow B\) and \(g: B \rightarrow C\) are two functions such that gof : \(A \rightarrow C\) is a bijections, then which one of the following is always true?
1 f and g are bijections
2 \(f\) is an injection and \(g\) is a surjection
3 \(f\) is a surjection and \(g\) is an injection
4 \(f\) is a bijection but \(g\) is not a bijection
Explanation:
B \(\operatorname{gof}(\mathrm{y})=\mathrm{z}\) \(g(f(y))=z\) \(f\) lies range but \(\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}\) for \(a, y \in A, f(y) \in B\), let \(f(y)=x\) For every element \(\mathrm{z} \in \mathrm{c}\), we have at \(\mathrm{x} \in \mathrm{B}\). Such that \(\mathrm{g}(\mathrm{x})=\mathrm{z}\) \(\therefore \quad \mathrm{g}\) is onto. Hence, \(\mathrm{f}\) is an injection and \(\mathrm{g}\) is a surjection.
Shift-I
Sets, Relation and Function
117229
Which of the following is true?
1 The composition of function is commutative
2 Every function is invertible
3 If a function \(f\) is bijective then its inverse \(\mathrm{f}^{-1}\) need not be bijective
4 The composition of functions is associative
Explanation:
D Consider \(\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}\) and \(\mathrm{g}: \mathrm{B} \rightarrow \mathrm{C}\) be two functions. Then the composition of \(f\) and \(g\) denoted by gof. is defined as the function gof: \(\mathrm{A} \rightarrow \mathrm{C}\) given by go gof \((x)=g f(x)\) Properties of function compositions. Associative property : Associative property of function composition, If there are three functions \(\mathrm{f}, \mathrm{g}\) and \(\mathrm{h}\), then they are said to be associative if and only is - \(\text { fo }(g o h)=(f o g) \text { oh. }\) The function composition of two onto function is always onto. The function composition of one - to - one function is always onto. The function composition of one- to-one function is always one to one So, the composition of function is associative is true statement.
JCECE-2018
Sets, Relation and Function
117235
If \(f(x)=\frac{4 x+7}{7 x-4}\), then the value of \(\mathbf{f}\{\mathbf{f}[\mathbf{f}(\mathbf{2})]\}=\)
117227
If \(f: A \rightarrow B\) and \(g: B \rightarrow C\) are two functions such that gof : \(A \rightarrow C\) is a bijections, then which one of the following is always true?
1 f and g are bijections
2 \(f\) is an injection and \(g\) is a surjection
3 \(f\) is a surjection and \(g\) is an injection
4 \(f\) is a bijection but \(g\) is not a bijection
Explanation:
B \(\operatorname{gof}(\mathrm{y})=\mathrm{z}\) \(g(f(y))=z\) \(f\) lies range but \(\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}\) for \(a, y \in A, f(y) \in B\), let \(f(y)=x\) For every element \(\mathrm{z} \in \mathrm{c}\), we have at \(\mathrm{x} \in \mathrm{B}\). Such that \(\mathrm{g}(\mathrm{x})=\mathrm{z}\) \(\therefore \quad \mathrm{g}\) is onto. Hence, \(\mathrm{f}\) is an injection and \(\mathrm{g}\) is a surjection.
Shift-I
Sets, Relation and Function
117229
Which of the following is true?
1 The composition of function is commutative
2 Every function is invertible
3 If a function \(f\) is bijective then its inverse \(\mathrm{f}^{-1}\) need not be bijective
4 The composition of functions is associative
Explanation:
D Consider \(\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}\) and \(\mathrm{g}: \mathrm{B} \rightarrow \mathrm{C}\) be two functions. Then the composition of \(f\) and \(g\) denoted by gof. is defined as the function gof: \(\mathrm{A} \rightarrow \mathrm{C}\) given by go gof \((x)=g f(x)\) Properties of function compositions. Associative property : Associative property of function composition, If there are three functions \(\mathrm{f}, \mathrm{g}\) and \(\mathrm{h}\), then they are said to be associative if and only is - \(\text { fo }(g o h)=(f o g) \text { oh. }\) The function composition of two onto function is always onto. The function composition of one - to - one function is always onto. The function composition of one- to-one function is always one to one So, the composition of function is associative is true statement.
JCECE-2018
Sets, Relation and Function
117235
If \(f(x)=\frac{4 x+7}{7 x-4}\), then the value of \(\mathbf{f}\{\mathbf{f}[\mathbf{f}(\mathbf{2})]\}=\)
117227
If \(f: A \rightarrow B\) and \(g: B \rightarrow C\) are two functions such that gof : \(A \rightarrow C\) is a bijections, then which one of the following is always true?
1 f and g are bijections
2 \(f\) is an injection and \(g\) is a surjection
3 \(f\) is a surjection and \(g\) is an injection
4 \(f\) is a bijection but \(g\) is not a bijection
Explanation:
B \(\operatorname{gof}(\mathrm{y})=\mathrm{z}\) \(g(f(y))=z\) \(f\) lies range but \(\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}\) for \(a, y \in A, f(y) \in B\), let \(f(y)=x\) For every element \(\mathrm{z} \in \mathrm{c}\), we have at \(\mathrm{x} \in \mathrm{B}\). Such that \(\mathrm{g}(\mathrm{x})=\mathrm{z}\) \(\therefore \quad \mathrm{g}\) is onto. Hence, \(\mathrm{f}\) is an injection and \(\mathrm{g}\) is a surjection.
Shift-I
Sets, Relation and Function
117229
Which of the following is true?
1 The composition of function is commutative
2 Every function is invertible
3 If a function \(f\) is bijective then its inverse \(\mathrm{f}^{-1}\) need not be bijective
4 The composition of functions is associative
Explanation:
D Consider \(\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}\) and \(\mathrm{g}: \mathrm{B} \rightarrow \mathrm{C}\) be two functions. Then the composition of \(f\) and \(g\) denoted by gof. is defined as the function gof: \(\mathrm{A} \rightarrow \mathrm{C}\) given by go gof \((x)=g f(x)\) Properties of function compositions. Associative property : Associative property of function composition, If there are three functions \(\mathrm{f}, \mathrm{g}\) and \(\mathrm{h}\), then they are said to be associative if and only is - \(\text { fo }(g o h)=(f o g) \text { oh. }\) The function composition of two onto function is always onto. The function composition of one - to - one function is always onto. The function composition of one- to-one function is always one to one So, the composition of function is associative is true statement.
JCECE-2018
Sets, Relation and Function
117235
If \(f(x)=\frac{4 x+7}{7 x-4}\), then the value of \(\mathbf{f}\{\mathbf{f}[\mathbf{f}(\mathbf{2})]\}=\)
