117267
Let \(f(x+y)=f(x) f(y)\) for all \(x\) and \(y\). If \(f(0)=1\), \(f(3)=3\) and \(f^{\prime}(0)=11\), then \(f^{\prime}(3)\) is equal to
1 11
2 22
3 33
4 44
5 55
Explanation:
C Given, \(f(x+y)=f(x) \cdot f(y) \text { and } f(3)=3, f^{\prime}(0)=11\) \(f(x+y)=f(x) \cdot f(y)\) \(f^{\prime}(x+y)=f(x) \cdot f^{\prime}(y)\) \(\text { put } x=3 \text { and } y=0 \text { in } \mathrm{eq}^n(i)\) \(f^{\prime}(3+0)=f(3) \cdot f^{\prime}(0)\) \(f^{\prime}(3)=3 \times 11 \quad \ldots(i)\)Hence, \(\mathrm{f}^{\prime}(3)=33\)
Kerala CEE-2017
Sets, Relation and Function
117269
If \(f(x)=3 x+5\) and \(g(x)=x^2-1\), then (fog) \(\left(x^2\right.\) -1) is equal to
1 \(3 x^4-3 x+5\)
2 \(3 x^4-6 x^2+5\)
3 \(6 x^4+3 x^2+5\)
4 \(6 x^4-6 x+5\)
5 \(3 x^2+6 x+4\)
Explanation:
B Given, \(f(x)=3 x+5\) And, \(g(x)=x^2-1\) Fog \(\left(\mathrm{x}^2-1\right)=\mathrm{f}\left(\mathrm{g}(\mathrm{x})^2-1\right)\) \(\therefore \quad \text { fog }\left(x^2-1\right) =f\left[g\left(x^2-1\right)\right]\) \(=f\left[\left(x^2-1\right)^2-1\right]\) \(=f\left[x^4-2 x^2+1-1\right]\) \(=f\left[x^4-2 x^2\right]\) \(=3\left(x^4-2 x^2\right)+5\)Hence, fog \(\left(x^2-1\right)=3 x^4-6 x^2+5\)
Kerala CEE-2016
Sets, Relation and Function
117271
If \(f(x)=\sin \mathrm{x}+\cos \mathrm{x}, \mathrm{x} \in(-\infty, \infty)\) and \(g(\mathrm{x})=\) \(x^2, x \in(-\infty, \infty)\) then (fog)(x) is equal to
117267
Let \(f(x+y)=f(x) f(y)\) for all \(x\) and \(y\). If \(f(0)=1\), \(f(3)=3\) and \(f^{\prime}(0)=11\), then \(f^{\prime}(3)\) is equal to
1 11
2 22
3 33
4 44
5 55
Explanation:
C Given, \(f(x+y)=f(x) \cdot f(y) \text { and } f(3)=3, f^{\prime}(0)=11\) \(f(x+y)=f(x) \cdot f(y)\) \(f^{\prime}(x+y)=f(x) \cdot f^{\prime}(y)\) \(\text { put } x=3 \text { and } y=0 \text { in } \mathrm{eq}^n(i)\) \(f^{\prime}(3+0)=f(3) \cdot f^{\prime}(0)\) \(f^{\prime}(3)=3 \times 11 \quad \ldots(i)\)Hence, \(\mathrm{f}^{\prime}(3)=33\)
Kerala CEE-2017
Sets, Relation and Function
117269
If \(f(x)=3 x+5\) and \(g(x)=x^2-1\), then (fog) \(\left(x^2\right.\) -1) is equal to
1 \(3 x^4-3 x+5\)
2 \(3 x^4-6 x^2+5\)
3 \(6 x^4+3 x^2+5\)
4 \(6 x^4-6 x+5\)
5 \(3 x^2+6 x+4\)
Explanation:
B Given, \(f(x)=3 x+5\) And, \(g(x)=x^2-1\) Fog \(\left(\mathrm{x}^2-1\right)=\mathrm{f}\left(\mathrm{g}(\mathrm{x})^2-1\right)\) \(\therefore \quad \text { fog }\left(x^2-1\right) =f\left[g\left(x^2-1\right)\right]\) \(=f\left[\left(x^2-1\right)^2-1\right]\) \(=f\left[x^4-2 x^2+1-1\right]\) \(=f\left[x^4-2 x^2\right]\) \(=3\left(x^4-2 x^2\right)+5\)Hence, fog \(\left(x^2-1\right)=3 x^4-6 x^2+5\)
Kerala CEE-2016
Sets, Relation and Function
117271
If \(f(x)=\sin \mathrm{x}+\cos \mathrm{x}, \mathrm{x} \in(-\infty, \infty)\) and \(g(\mathrm{x})=\) \(x^2, x \in(-\infty, \infty)\) then (fog)(x) is equal to
117267
Let \(f(x+y)=f(x) f(y)\) for all \(x\) and \(y\). If \(f(0)=1\), \(f(3)=3\) and \(f^{\prime}(0)=11\), then \(f^{\prime}(3)\) is equal to
1 11
2 22
3 33
4 44
5 55
Explanation:
C Given, \(f(x+y)=f(x) \cdot f(y) \text { and } f(3)=3, f^{\prime}(0)=11\) \(f(x+y)=f(x) \cdot f(y)\) \(f^{\prime}(x+y)=f(x) \cdot f^{\prime}(y)\) \(\text { put } x=3 \text { and } y=0 \text { in } \mathrm{eq}^n(i)\) \(f^{\prime}(3+0)=f(3) \cdot f^{\prime}(0)\) \(f^{\prime}(3)=3 \times 11 \quad \ldots(i)\)Hence, \(\mathrm{f}^{\prime}(3)=33\)
Kerala CEE-2017
Sets, Relation and Function
117269
If \(f(x)=3 x+5\) and \(g(x)=x^2-1\), then (fog) \(\left(x^2\right.\) -1) is equal to
1 \(3 x^4-3 x+5\)
2 \(3 x^4-6 x^2+5\)
3 \(6 x^4+3 x^2+5\)
4 \(6 x^4-6 x+5\)
5 \(3 x^2+6 x+4\)
Explanation:
B Given, \(f(x)=3 x+5\) And, \(g(x)=x^2-1\) Fog \(\left(\mathrm{x}^2-1\right)=\mathrm{f}\left(\mathrm{g}(\mathrm{x})^2-1\right)\) \(\therefore \quad \text { fog }\left(x^2-1\right) =f\left[g\left(x^2-1\right)\right]\) \(=f\left[\left(x^2-1\right)^2-1\right]\) \(=f\left[x^4-2 x^2+1-1\right]\) \(=f\left[x^4-2 x^2\right]\) \(=3\left(x^4-2 x^2\right)+5\)Hence, fog \(\left(x^2-1\right)=3 x^4-6 x^2+5\)
Kerala CEE-2016
Sets, Relation and Function
117271
If \(f(x)=\sin \mathrm{x}+\cos \mathrm{x}, \mathrm{x} \in(-\infty, \infty)\) and \(g(\mathrm{x})=\) \(x^2, x \in(-\infty, \infty)\) then (fog)(x) is equal to
NEET Test Series from KOTA - 10 Papers In MS WORD
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Sets, Relation and Function
117267
Let \(f(x+y)=f(x) f(y)\) for all \(x\) and \(y\). If \(f(0)=1\), \(f(3)=3\) and \(f^{\prime}(0)=11\), then \(f^{\prime}(3)\) is equal to
1 11
2 22
3 33
4 44
5 55
Explanation:
C Given, \(f(x+y)=f(x) \cdot f(y) \text { and } f(3)=3, f^{\prime}(0)=11\) \(f(x+y)=f(x) \cdot f(y)\) \(f^{\prime}(x+y)=f(x) \cdot f^{\prime}(y)\) \(\text { put } x=3 \text { and } y=0 \text { in } \mathrm{eq}^n(i)\) \(f^{\prime}(3+0)=f(3) \cdot f^{\prime}(0)\) \(f^{\prime}(3)=3 \times 11 \quad \ldots(i)\)Hence, \(\mathrm{f}^{\prime}(3)=33\)
Kerala CEE-2017
Sets, Relation and Function
117269
If \(f(x)=3 x+5\) and \(g(x)=x^2-1\), then (fog) \(\left(x^2\right.\) -1) is equal to
1 \(3 x^4-3 x+5\)
2 \(3 x^4-6 x^2+5\)
3 \(6 x^4+3 x^2+5\)
4 \(6 x^4-6 x+5\)
5 \(3 x^2+6 x+4\)
Explanation:
B Given, \(f(x)=3 x+5\) And, \(g(x)=x^2-1\) Fog \(\left(\mathrm{x}^2-1\right)=\mathrm{f}\left(\mathrm{g}(\mathrm{x})^2-1\right)\) \(\therefore \quad \text { fog }\left(x^2-1\right) =f\left[g\left(x^2-1\right)\right]\) \(=f\left[\left(x^2-1\right)^2-1\right]\) \(=f\left[x^4-2 x^2+1-1\right]\) \(=f\left[x^4-2 x^2\right]\) \(=3\left(x^4-2 x^2\right)+5\)Hence, fog \(\left(x^2-1\right)=3 x^4-6 x^2+5\)
Kerala CEE-2016
Sets, Relation and Function
117271
If \(f(x)=\sin \mathrm{x}+\cos \mathrm{x}, \mathrm{x} \in(-\infty, \infty)\) and \(g(\mathrm{x})=\) \(x^2, x \in(-\infty, \infty)\) then (fog)(x) is equal to
117267
Let \(f(x+y)=f(x) f(y)\) for all \(x\) and \(y\). If \(f(0)=1\), \(f(3)=3\) and \(f^{\prime}(0)=11\), then \(f^{\prime}(3)\) is equal to
1 11
2 22
3 33
4 44
5 55
Explanation:
C Given, \(f(x+y)=f(x) \cdot f(y) \text { and } f(3)=3, f^{\prime}(0)=11\) \(f(x+y)=f(x) \cdot f(y)\) \(f^{\prime}(x+y)=f(x) \cdot f^{\prime}(y)\) \(\text { put } x=3 \text { and } y=0 \text { in } \mathrm{eq}^n(i)\) \(f^{\prime}(3+0)=f(3) \cdot f^{\prime}(0)\) \(f^{\prime}(3)=3 \times 11 \quad \ldots(i)\)Hence, \(\mathrm{f}^{\prime}(3)=33\)
Kerala CEE-2017
Sets, Relation and Function
117269
If \(f(x)=3 x+5\) and \(g(x)=x^2-1\), then (fog) \(\left(x^2\right.\) -1) is equal to
1 \(3 x^4-3 x+5\)
2 \(3 x^4-6 x^2+5\)
3 \(6 x^4+3 x^2+5\)
4 \(6 x^4-6 x+5\)
5 \(3 x^2+6 x+4\)
Explanation:
B Given, \(f(x)=3 x+5\) And, \(g(x)=x^2-1\) Fog \(\left(\mathrm{x}^2-1\right)=\mathrm{f}\left(\mathrm{g}(\mathrm{x})^2-1\right)\) \(\therefore \quad \text { fog }\left(x^2-1\right) =f\left[g\left(x^2-1\right)\right]\) \(=f\left[\left(x^2-1\right)^2-1\right]\) \(=f\left[x^4-2 x^2+1-1\right]\) \(=f\left[x^4-2 x^2\right]\) \(=3\left(x^4-2 x^2\right)+5\)Hence, fog \(\left(x^2-1\right)=3 x^4-6 x^2+5\)
Kerala CEE-2016
Sets, Relation and Function
117271
If \(f(x)=\sin \mathrm{x}+\cos \mathrm{x}, \mathrm{x} \in(-\infty, \infty)\) and \(g(\mathrm{x})=\) \(x^2, x \in(-\infty, \infty)\) then (fog)(x) is equal to