Solving the Functional Equation
We are given the functional equation: $f (x)+3f (\frac{24}{x}) = 4x$, where $x \neq 0$. We need to find the value of $f (3) + f (8)$.
Evaluating the Function at Specific Points
Substitute $x = 3$ into the equation:
$f (3)+3f (\frac{24}{3}) = 4(3)$
$f (3)+3f (8) = 12 \quad \cdots (1)$
Substitute $x = 8$ into the equation:
$f (8)+3f (\frac{24}{8}) = 4(8)$
$f (8)+3f (3) = 32 \quad \cdots (2)$
Solving the System of Equations
We now have a system of two linear equations with two variables, $f(3)$ and $f(8)$:
- $f(3) + 3f(8) = 12$
- $3f(3) + f(8) = 32$
Multiply Equation (1) by 3:
$3(f(3) + 3f(8)) = 3(12)$
$3f(3) + 9f(8) = 36 \quad \cdots (3)$
Subtract Equation (2) from Equation (3):
$(3f(3) + 9f(8)) - (3f(3) + f(8)) = 36 - 32$
$8f(8) = 4
$f(8) = \frac{4}{8} = \frac{1}{2}
Substitute the value of $f(8)$ back into Equation (1):
$f(3) + 3(\frac{1}{2}) = 12$
$f(3) + \frac{3}{2} = 12$
$f(3) = 12 - \frac{3}{2} = \frac{24}{2} - \frac{3}{2} = \frac{21}{2}
Calculating the Final Value
We need to find $f(3) + f(8)$.
$f(3) + f(8) = \frac{21}{2} + \frac{1}{2} = \frac{22}{2} = 11
Similar Questions
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $P = [p_{ij}]$ and $Q = [q_{ij}]$ be two square matrices of order 3 such that $q_{ij} = 2^{(i + j - 1)} p_{ij}$ and $\det(Q) = 2^{10}$. Then the value of $\det(\text{adj}(\text{adj } P))$ is:
- The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is
- Let $f$ be a function such that $3f(x) + 2f\left(\frac{m}{19x}\right) = 5x, x \neq 0$, where $m = \sum_{i=1}^{9} (i)^2$. Then $f(5) - f(2)$ is equal to
- The smallest positive integral value of $a$, for which all the roots of $x^4 - ax^2 + 9 = 0$ are real and distinct, is equal to
- $\left(\frac{1}{3} + \frac{4}{7}\right) + \left(\frac{1}{3^2} + \frac{1}{3} \times \frac{4}{7} + \frac{4^2}{7^2}\right) + \left(\frac{1}{3^3} + \frac{1}{3^2} \times \frac{4}{7} + \frac{1}{3} \times \frac{4^2}{7^2} + \frac{4^3}{7^3}\right) + \dots$ upto infinite terms, is equal to
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
Important Questions from Algebra
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $P = [p_{ij}]$ and $Q = [q_{ij}]$ be two square matrices of order 3 such that $q_{ij} = 2^{(i + j - 1)} p_{ij}$ and $\det(Q) = 2^{10}$. Then the value of $\det(\text{adj}(\text{adj } P))$ is:
