Consider f : R+ → [−5, ∞) given by f(x) = 9x2 + 6x − 5. Show that f is invertible with
Question: Consider $f: R_{+} \rightarrow[-5, \infty)$ given by $f(x)=9 x^{2}+6 x-5$. Show that $f$ is invertible with $f^{-1}(x)=\frac{\sqrt{x+6}-1}{3}$. Solution: Injectivity of $f$ : Let $x$ and $y$ be two elements of domain $\left(R^{+}\right)$, such that $f(x)=f(y)$ $\Rightarrow 9 x^{2}+6 x-5=9 y^{2}+6 y-5$ $\Rightarrow 9 x^{2}+6 x=9 y^{2}+6 y$ $\Rightarrow x=y\left(\right.$ As $\left., x, y \in R^{+}\right)$ So, $f$ is one-one. Surjectivity off:Letyis in the co domain (Q) such thatf(x) = y ...
Read More →Factorize:
Question: Factorize: $2 x^{2}+11 x-21$ Solution: We have: $2 x^{2}+11 x-21$ We have to split 11 into two numbers such that their sum is 11 and their product is $(-42)$, i.e., $2 \times(-21)$. Clearly, $14+(-3)=11$ and $14 \times(-3)=-42$. $\therefore 2 x^{2}+11 x-21=2 x^{2}+14 x-3 x-21$ $=2 x(x+7)-3(x+7)$ $=(x+7)(2 x-3)$...
Read More →Divide 29 into two parts so that the sum of the squares of the parts is 425.
Question: Divide 29 into two parts so that the sum of the squares of the parts is 425. Solution: Let first numbers be $x$ and other $(29-x)$ Then according to question $x^{2}+(29-x)^{2}=425$ $x^{2}+x^{2}-58 x+841=425$ $2 x^{2}-58 x+841=425$ $2 x^{2}-58 x+841-425=0$ $2 x^{2}-58 x+416=0$ $x^{2}-29 x+208=0$ $x^{2}-16 x-13 x+208=0$ $x(x-16)-13(x-16)=0$ $(x-16)(x+13)=0$ $(x-16)=0$ $x=16$ Or $(x+13)=0$ $x=-13$ Since, 29being a positive number, soxcannot be negative. Therefore, When $x=16$ then $29-x=2...
Read More →Solve each of the following system of equations in R. 20. −5 < 2x − 3 < 5
Question: Solve each of the following system of equations in R. 20. 5 2x 3 5 Solution: $-52 x-35$ $\Rightarrow-5+32 x5+3 \quad$ (Adding 3 throughout) $\Rightarrow-22 x8$ $\Rightarrow-1x4 \quad$ (Dividing throughout by 2 ) $\Rightarrow x \in(-1,4)$ Hence, the interval (-1,4) is the solution of the given set of inequaltions....
Read More →Solve each of the following system of equations in R. 19. 10 ≤ −5 (x − 2) < 20
Question: Solve each of the following system of equations in R. 19. 10 5 (x 2) 20 Solution: $10 \leq-5(x-2)20$ $\Rightarrow 10 \leq-5 x+1020$ $\Rightarrow 10-10 \leq-5 x20-10 \quad$ (Substracting 10 from all three terms) $\Rightarrow 0 \leq-5 x10$ $\Rightarrow 0 \geq x-2 \quad$ (Dividing all three terms by $-5$ ) $\Rightarrow x \in(-2,0]$ Hence, the interval $(-2,0]$ is the solution of the given set of inequations....
Read More →Factorize:
Question: Factorize: $18 x^{2}+3 x-10$ Solution: We have: $18 x^{2}+3 x-10$ We have to split 3 into two numbers such that their sum is 3 and their product is $(-180)$, i.e., $18 \times(-10)$. Clearly, $15+(-12)=3$ and $15 \times(-12)=-180$ $\therefore 18 x^{2}+3 x-10=18 x^{2}+15 x-12 x-10$ $=3 x(6 x+5)-2(6 x+5)$ $=(6 x+5)(3 x-2)$...
Read More →If f(x)
Question: If $f(x)=\frac{4 x+3}{6 x-4}, x \neq \frac{2}{3}$, show that $f \circ f(x)=x$ for all $x \neq \frac{2}{3}$. What is the inverse of $f$ ? Solution: $(f o f)(x)=f(f(x))$ $=f\left(\frac{4 x+3}{6 x-4}\right)$ $=\frac{4\left(\frac{4 x+3}{6 x-4}\right)+3}{6\left(\frac{4 x+3}{6 x-4}\right)-4}$ $=\frac{16 x+12+18 x-12}{24 x+18-24 x+16}$ $=\frac{34 x}{34}$ $=x$ $\Rightarrow(f o f)(x)=x=I_{X}$, where $I$ is an identity function. So, $f=f^{-1}$ Hence, $f^{-1}=\frac{4 x+3}{6 x-4}$...
Read More →Solve each of the following system of equations in R. 18.
Question: Solve each of the following system of equations in R. 18. $0\frac{-x}{2}3$ Solution: $0\frac{-x}{2}3$ $\Rightarrow 0-x6 \quad$ (Multiplying throughout by 2 ) $\Rightarrow 0x-6 \quad$ (Multiplying throughout by $-1$ ) $\Rightarrow x \in(-6,0)$ Hence, the interval $(-6,0)$ is the solution of the given set of inequations....
Read More →Find two consecutive numbers whose squares have the sum 85.
Question: Find two consecutive numbers whose squares have the sum 85. Solution: Let two consecutive numbers be $x$ and $(x+1)$ Then according to question $x^{2}+(x+1)^{2}=85$ $x^{2}+x^{2}+2 x+1=85$ $2 x^{2}+2 x-85+1=0$ $2 x^{2}+2 x-84=0$ $x^{2}+x-42=0$ $x^{2}+7 x-6 x-42=0$ $x(x+7)-6(x+7)=0$ $(x+7)(x-6)=0$ $(x+7)=0$ $x=-7$ Or $(x-6)=0$ $x=6$ Since,xbeing a number, Therefore, When $x=-7$ then $x+1=-7+1$ $=-6$ And when $x=6$ then $x+1=6+1$ $=7$ Thus, two consecutive number be either 6,7 or $-6,-7$...
Read More →Solve each of the following system of equations in R.
Question: Solve each of the following system of equations in R. 17. $\frac{2 x+1}{7 x-1}5, \frac{x+7}{x-8}2$ Solution: $\frac{2 x+1}{7 x-1}5$ $\Rightarrow \frac{2 x+1}{7 x-1}-50$ $\Rightarrow \frac{2 x+1-35 x+5}{7 x-1}0$ $\Rightarrow \frac{6-33 x}{7 x-1}0$ $\Rightarrow x \in\left(\frac{1}{7}, \frac{2}{11}\right)$ ...(i) Also, $\frac{x+7}{x-8}2$ $\Rightarrow \frac{x+7}{x-8}-20$ $\Rightarrow \frac{x+7-2 x+16}{x-8}0$ $\Rightarrow \frac{23-x}{x-8}0$ $\Rightarrow x \in(-\infty, 8) \cup(23, \infty) \q...
Read More →Factorise:
Question: Factorise: $6 x^{2}+17 x+12$ Solution: $6 x^{2}+17 x+12$ $=6 x^{2}+9 x+8 x+12$ $=3 x(2 x+3)+4(2 x+3)$ $=(2 x+3)(3 x+4)$...
Read More →Consider f : R → R+ → [4, ∞) given by f(x) = x2 + 4.
Question: Consider $f: R \rightarrow R_{+} \rightarrow[4, \infty)$ given by $f(x)=x^{2}+4$. Show that $f$ is invertible with inverse $f^{-1}$ of $f$ given by $f^{-1}(x)=\sqrt{x-4}$, where $R^{+}$is the set of all non-negative real numbers. Solution: Injectivity of $f$ : Let $x$ and $y$ be two elements of the domain $(Q)$, such that $f(x)=f(y)$ $\Rightarrow x^{2}+4=y^{2}+4$ $\Rightarrow x^{2}=y^{2}$ $\Rightarrow x=y \quad\left(\right.$ as co-domain as $\left.R^{+}\right)$ So, $f$ is one-one. Surj...
Read More →Determine. if 3 is a root of the equation given below:
Question: Determine. if 3 is a root of the equation given below: $\sqrt{x^{2}-4 x+3}+\sqrt{x^{2}-9}=\sqrt{4 x^{2}-14 x+16}$ Solution: We have been given that, $\sqrt{x^{2}-4 x+3}+\sqrt{x^{2}-9}=\sqrt{4 x^{2}-14 x+16}$ We have to check whetherx= 3 is the solution of the given equation or not. Now, if $x=3$ is a root of the above quadratic equation, then it should satisfy the whole. So substituting $x=3$ in the above equation, we have, Left hand side $=\sqrt{(3)^{2}-4(3)+3}+\sqrt{(3)^{2}-9}$ $=\sq...
Read More →Solve each of the following system of equations in R.
Question: Solve each of the following system of equations in R. 16. $\frac{7 x-1}{2}-3, \frac{3 x+8}{5}+110$ Solution: $\frac{7 x-1}{2}-3$ $\Rightarrow 7 x-1-6$ $\Rightarrow 7 x-6+1$ $\Rightarrow x\frac{-5}{7}$ $\Rightarrow x \in\left(-\infty, \frac{-5}{7}\right) \quad \ldots$ (i) $\Rightarrow \frac{3 x+8+55}{5}0$ $\Rightarrow 3 x+630$ $\Rightarrow 3 x-63$ $\Rightarrow x-21$ $\Rightarrow x \in(-\infty,-21) \quad \ldots$ (ii) Hence, the solution to the given set of inequations is the intersection...
Read More →Factorise:
Question: Factorise: $9 x^{2}+18 x+8$ Solution: $9 x^{2}+18 x+8$ $=9 x^{2}+12 x+6 x+8$ $=3 x(3 x+4)+2(3 x+4)$ $=(3 x+4)(3 x+2)$...
Read More →Factorise:
Question: Factorise: $x^{2}-32 x-105$ Solution: $x^{2}-32 x-105$ $=x^{2}-35 x+3 x-105$ $=x(x-35)+3(x-35)$ $=(x-35)(x+3)$...
Read More →Consider f : R → R given by f(x) = 4x + 3.
Question: Consider $f: R \rightarrow R$ given by $f(x)=4 x+3$. Show that $f$ is invertible. Find the inverse of $f$. Solution: Injectivity of $f$ : Let $x$ and $y$ be two elements of domain $(R)$, such that $f(x)=f(y)$ $\Rightarrow 4 x+3=4 y+3$ $\Rightarrow 4 x=4 y$ $\Rightarrow x=y$ So, $f$ is one-one. Surjectivity of $f$ : Let $y$ be in the co-domain $(R)$, such that $f(x)=y$. $\Rightarrow 4 x+3=y$ $\Rightarrow 4 x=y-3$ $\Rightarrow x=\frac{y-3}{4} \in R($ Domain $)$ $\Rightarrow f$ is onto. S...
Read More →If x = 2/3 and x = −3 are the roots of the equation
Question: If $x=2 / 3$ and $x=-3$ are the roots of the equation $a x^{2}+7 x+b=0$, find the values of $a$ and $b$. Solution: We have been given that, $a x^{2}+7 x+b=0, x=\frac{2}{3}, x=-3$ We have to findaandb Now, if $x=\frac{2}{3}$ is a root of the equation, then it should satisfy the equation completely. Therefore we substitute $x=\frac{2}{3}$ in the above equation. We get, $a\left(\frac{2}{3}\right)^{2}+7\left(\frac{2}{3}\right)+b=0$ $\frac{4 a+42+9 b}{9}=0$ $a=\frac{-9 b-42}{4} \ldots \ldot...
Read More →Solve each of the following system of equations in R.
Question: Solve each of the following system of equations in R. 15. $\frac{2 x-3}{4}-2 \geq \frac{4 x}{3}-6,2(2 x+3)6(x-2)+10$ Solution: $\frac{2 x-3}{4}-2 \geq \frac{4 x}{3}-6$ $\Rightarrow \frac{2 x-3}{4}-\frac{4 x}{3} \geq-6+2$ $\Rightarrow \frac{3(2 x-3)-16 x}{12} \geq-4$ $\Rightarrow 6 x-9-16 x \geq-48$ $\Rightarrow-10 x \geq-39$ $\Rightarrow 10 x \leq 39 \quad$ [Multiplying both sides by $-1]$ $\Rightarrow x \leq \frac{39}{10}$ $\Rightarrow x \in\left(-\infty, \frac{39}{10}\right] \quad \l...
Read More →Factorize:
Question: Factorize: $x^{2}-x-156$ Solution: We have: $x^{2}-x-156$ We have to split (-1) into two numbers such that their sum is (-1) and their product is (-156). Clearly, $-13+12=-1$ and $-13 \times 12=-156$ $\therefore x^{2}-x-156=x^{2}-13 x+12 x-156$ $=x(x-13)+12(x-13)$ $=(x-13)(x+12)$...
Read More →Factorise:
Question: Factorise: $x^{2}-2 \sqrt{2} x-30$ Solution: $x^{2}-2 \sqrt{2} x-30$ $=x^{2}-5 \sqrt{2} x+3 \sqrt{2} x-30$ $=x(x-5 \sqrt{2})+3 \sqrt{2}(x-5 \sqrt{2})$ $=(x-5 \sqrt{2})(x+3 \sqrt{2})$...
Read More →Show that the function f : Q → Q, defined by f(x) = 3x + 5,
Question: Show that the function $f: Q \rightarrow Q$, defined by $f(x)=3 x+5$, is invertible. Also, find $f^{-1}$ Solution: Injectivity of $f$ : Let $x$ and $y$ be two elements of the domain $(Q)$, such that $f(x)=f(y)$ $\Rightarrow 3 x+5=3 y+5$ $\Rightarrow 3 x=3 y$ $\Rightarrow x=y$ So, $f$ is one-one. Surjectivity of $f$. Let $y$ be in the co-domain $(Q)$, such that $f(x)=y$ $\Rightarrow 3 x+5=y$ $\Rightarrow 3 x=y-5$ $\Rightarrow x=\frac{y-5}{3} \in Q($ domain $)$ $\Rightarrow f$ is onto. S...
Read More →Factorise:
Question: Factorise: $x^{2}+\sqrt{2} x-24$ Solution: $x^{2}+\sqrt{2} x-24$ $=x^{2}+4 \sqrt{2} x-3 \sqrt{2} x-24$ $=x(x+4 \sqrt{2})-3 \sqrt{2}(x+4 \sqrt{2})$ $=(x+4 \sqrt{2})(x-3 \sqrt{2})$...
Read More →In each of the following, find the value of k for which the given value is a solution of the given equation:
Question: In each of the following, find the value ofkfor which the given value is a solution of the given equation: (i) $7 x^{2}+k x-3=0, x=\frac{2}{3}$ (ii) $x^{2}-x(a+b)+k=0, x=a$ (iii) $k x^{2}+\sqrt{2} x-4=0, x=\sqrt{2}$ (iv) $x^{2}+3 a x+k=0, x=-a$ Solution: In each of the following cases findk. (i) We are given here that, $7 x^{2}+k x-3=0, x=\frac{2}{3}$ Now, as we know that $x=\frac{2}{3}$ is a solution of the quadratic equation, hence it should satisfy the equation. Therefore substituti...
Read More →Solve each of the following system of equations in R.
Question: Solve each of the following system of equations in R. 14. $5 x-73(x+3), 1-\frac{3 x}{2} \geq x-4$ Solution: $5 x-73(x+3)$ $\Rightarrow 5 x-73 x+9$ $\Rightarrow 5 x-3 x9+7$ $\Rightarrow 2 x16$ $\Rightarrow x8$ $\Rightarrow x \in(-\infty, 8) \quad \ldots(\mathrm{i})$ Also, $1-\frac{3 x}{2} \geq x-4$ $\Rightarrow x-4 \leq 1-\frac{3 x}{2}$ $\Rightarrow x+\frac{3 x}{2} \leq 1+4$ $\Rightarrow \frac{2 x+3 x}{2} \leq 5$ $\Rightarrow 5 x \leq 10$ $\Rightarrow x \leq 2$ $\Rightarrow x \in(-\inft...
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