Let $f, g$ and $h$ be functions from $R$ to $R$. Show that:
$\begin{cases} (f+g)oh = foh + goh \\ (f \cdot g)oh = (foh) \cdot (goh) \end{cases}$
$\begin{cases} (f+g)oh = foh + goh \\ (f \cdot g)oh = (foh) \cdot (goh) \end{cases}$
To prove: $(f+g)oh = foh + goh$
$LHS = [(f+g)oh](x)$
$= (f+g)[h(x)] = f[h(x)] + g[h(x)]$
$= (foh)(x) + (goh)(x)$
$= \{(foh) + (goh)\}(x) = RHS$
$\therefore \{(f+g)oh\}(x) = \{(foh) + (goh)\}(x)$ for all $x \in R$.
Hence,$(f+g)oh = foh + goh$.
To prove: $(f \cdot g)oh = (foh) \cdot (goh)$
$LHS = [(f \cdot g)oh](x)$
$= (f \cdot g)[h(x)] = f[h(x)] \cdot g[h(x)]$
$= (foh)(x) \cdot (goh)(x)$
$= \{(foh) \cdot (goh)\}(x) = RHS$
$\therefore [(f \cdot g)oh](x) = \{(foh) \cdot (goh)\}(x)$ for all $x \in R$.
Hence,$(f \cdot g)oh = (foh) \cdot (goh)$.
$LHS = [(f+g)oh](x)$
$= (f+g)[h(x)] = f[h(x)] + g[h(x)]$
$= (foh)(x) + (goh)(x)$
$= \{(foh) + (goh)\}(x) = RHS$
$\therefore \{(f+g)oh\}(x) = \{(foh) + (goh)\}(x)$ for all $x \in R$.
Hence,$(f+g)oh = foh + goh$.
To prove: $(f \cdot g)oh = (foh) \cdot (goh)$
$LHS = [(f \cdot g)oh](x)$
$= (f \cdot g)[h(x)] = f[h(x)] \cdot g[h(x)]$
$= (foh)(x) \cdot (goh)(x)$
$= \{(foh) \cdot (goh)\}(x) = RHS$
$\therefore [(f \cdot g)oh](x) = \{(foh) \cdot (goh)\}(x)$ for all $x \in R$.
Hence,$(f \cdot g)oh = (foh) \cdot (goh)$.
Explore More
Similar Questions
If $f(x) = \frac{x}{2x+1}$ and $g(x) = \frac{x}{x+1}$,then $(f \circ g)(x) = $
Let $f(x) = \log_e x$ and $g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1}$. Then the domain of $f \circ g$ is:
Let $f(x) = ax + b$ and $g(x) = cx + d$,where $a \ne 0$ and $c \ne 0$. Assume $a = 1$ and $b = 2$. If $(fog)(x) = (gof)(x)$ for all $x$,what can you say about $c$ and $d$?
Medium
View SolutionLet $f(x) = \frac{x+1}{x-1}$ for all $x \neq 1$. Let $f^1(x) = f(x)$,$f^2(x) = f(f(x))$ and generally $f^n(x) = f(f^{n-1}(x))$ for $n > 1$. Let $P = f^1(2) \cdot f^2(3) \cdot f^3(4) \cdot f^4(5)$. Which of the following is a multiple of $P$?
Let $f(x) = \sin x$ and $g(x) = \cos x$. Which of the following statements is false?
Vedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo