Let x2+kx=0;kx^2+kx=0;k is a real number .The set of values of kk for which the equation f(x)=0f(x)=0 and f(f(x))=0f(f(x))=0 have same real solution set.

Let f(x)=x2+kx;k is a real number.The set of values of k for which the equation f(x)=0 and f(f(x))=0 have same real solution set. The equation x2+kx=0 has solutions x=0,−k. So the solutions of the equation f(f(x))=0 should be x=0,−k. f(f(x))=(x2+kx)2+(x2+kx)k=0 gives x4+k2x2+2kx3+x2k+xk2=0 x4+2kx3+(k2+k)x2+xk2=0 x(x3+2kx2+(k2+k)x+k2)=0 has the solutions x=0,x=−k Therefore x3+2kx2+(k2+k)x+k2=0 has the solution x=−k. But … Read more

Can the composition of a rational function ever be a polynomial?

Suppose we have a function f(x)=h(x)p(x) where h(x) and p(x) are polynomials. Can any iterative composition of f(x) like f(f(x)) or f(f(…f(x)…) be a polynomial? Update Let me add a few more constraints. deg(p(x)),deg(h(x))≥1 and p(x)∤ Furthermore if \dfrac{h(x)}{p(x)} can be reduced, then neither the numerator nor the denominator is ever a constant, i.e the … Read more

What is composition of convex and concave function?

Suppose that f:R→R is convex function and g:R→R is concave function. What can we say about their composition g∘f and f∘g? Are they convex or concave functions? Thank you in advace ! Answer Hint. Try f(x)=ex (convex) and g(x)=−x2 (concave). What about f(g(x))=e−x2? Is it convex or concave? Check the plot at WA. P. S. … Read more

Composition of multiplicative functions which is not multiplicative

A function $f\colon\mathbb N\to\mathbb C$ is called multiplicative if $f(1)=1$ and $$\gcd(a,b)=1 \implies f(ab)=f(a)f(b).$$ It is called completely multiplicative if the equality $f(ab)=f(a)f(b)$ holds for any pair of positive integers $a$, $b$. (In the definition of multiplicative function we have this condition only for $a$, $b$ coprime.) If is not difficult to show that if … Read more

Finding g(x)g(x) , given h(x)h(x) and f(x)f(x)

find g(x) f(g(x))=h(x) f(x)=x2−6x+2 h(x)=9×2−24x+9 My work: g(x)2−6(g(x))+2=h(x) g(x)2−6(g(x))−(9×2−24x+7) And then what’s next step?please provide explaination as well Thank You!! 3+- sqr(9-(-9x^2-24x+7)) 3+- sqr(9+9x^2+24x-7) Then what do i do? Answer Note f(x)=(x−3)2−7h(x)=(3x−4)2−7 It follows that f(3x−1)=h(x). Hope this helps. AttributionSource : Link , Question Author : Secret , Answer Author : awllower

Use a composition of functions to define 14×4−12×3+9×2\frac{1}{4x^4-12x^3+9x^2}

14×4−12×3+9×2 The functions available are f(x)=2x g(x)=x−1 h(x)=x2 i(x)=2×2−3x j(x)=√x k(x)=1x I can only get as far as k(i(x)) and not really sure how to even go from there. Any solutions or help appreciated. 🙂 Answer 14×4−12×3+9×2=k(4×4−12×3+9×2) =k(h(2×2−3x))=k(h(i(x)) AttributionSource : Link , Question Author : Percocets , Answer Author : Parcly Taxel

How to solve for $y$ in the equation $e^y-ey=e^x-ex$?

can I solve for $y$ the equation $$e^y-ey=e^x-ex$$ with a simple trick nice and fast? If not how can I solve this equation with the simplest algebra possible? Actually I have the function $$f(x)=e^x-ex$$ and I want to find two other functions $g(x)$ and $h(x)$ in order the composition of them equals with the $f(x)$ … Read more

Prove any function can be written as a composition between an injective and a surjective function.

Given an arbitrary function f:A→B, write it as a composition between an injective and a surjective function, respectively. Answer This one is pretty straightforward. Intuitively, all we have to do is pick a function that maps set A to the image of your initial function, and a function that “extends” the image of f to … Read more

Iterations of $f(x)=\dfrac{ax+b}{cx+d}$

Consider $f(x)=\dfrac{ax+b}{cx+d}$, where $c\neq0$ and $f(x)$ is not equal to a constant. Is it necessarily true that $f^{[n]}(x)=f(x)$ for some natural number $n > 1$? Answer Consider $f(x)=\frac{x}{x+1}$. Then $f^{[n]}(x)=\frac{x}{nx+1}$. If this is true for some $n$ (and it is for $n=1$) then $$\begin{align}f^{[n+1]}(x)&=\frac{\frac{x}{x+1}}{n\frac{x}{x+1}+1}\\ &=\frac{x}{nx+(x+1)}\\ &=\frac{x}{(n+1)x+1}\end{align}$$ And so by induction $f^{[n]}(x)=\frac{x}{nx+1}$ for all $n\in\mathbb{N}$. This … Read more

Prove that if f∘g=f∘hf \circ g = f\circ h, then g=hg = h

Given three functions, f:A→B,g:C→A,h:C→A Prove or disprove that if f∘g=f∘h, then g=h. Answer Here is a counterexample: g(1)=1g(2)=2g(3)=3h(1)=1h(2)=3h(3)=2f(1)=1f(2)=2f(3)=2 AttributionSource : Link , Question Author : becozlah , Answer Author : Michael Hardy