Function functioned my brain

I was working with this function problem but, I got stuck. Here's the problem: Let f(x) be a function with the two properties_ a. For any two real numbers x and y, f(xy)=x.f(y) and _b. f(1)=25. What is the value of f(79)?

#Algebra #MathProblem #Math

Easy Math Editor

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Comments

Given $f(xy) = x\cdot f(y)$ and $f(1) = 25$ ,

Put $y = 1$ , we get $f(x) = x\cdot f(1) = 25x\Rightarrow f(x) = 25x$

So Put $x = 79$ , we get $f(79) = 25\cdot 79 = 1975$

jagdish singh - 7 years, 6 months ago

Hi, this seems simple:

$\text{f(xy) = xf(y)}$

Derivate with respect to $y$ taking $x$ constant to get:

$\text{xf'(xy) = xf'(y)}$

$\Rightarrow \text{f'(xy) = f'(x) = k}$ (say) Integrate to get $\text{f(x) = kx + c}$

$\text{f(xy) = x f(y)} \Rightarrow \text{kxy + c = kxy + xc}$ $\forall$ $x \in R$

Hence, $\text{c=0}$ .

Now, $\text{f(x) = kx}$ , and $f(1) = 25 \Rightarrow \text{f(x) = 25x}$

Hence, $\text{f(79) = 1975}$

jatin yadav - 7 years, 6 months ago

How do you know that you can differentiate this function? Why must it even be continuous?

Calvin Lin Staff - 7 years, 6 months ago

f(xy)=x.f(y) given now put y=1 f(x1)=xf(1) =25x (as f(1)=25) so now f(79)=25*79 =1975

Rohit Goyal - 7 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$