Let f(x) = ax + b and g(x) = cx + d,where a n | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given $f(x) = ax + b$ and $g(x) = cx + d$. We are given $a = 1$ and $b = 2$,so $f(x) = x + 2$. Now,calculate the composition $(fog)(x) = f(g(x)) =

Vedclass · Accepted Answer

$c = 1$,$d$ is arbitrary

Vedclass · Answer

(B) Given $f(x) = ax + b$ and $g(x) = cx + d$. We are given $a = 1$ and $b = 2$,so $f(x) = x + 2$. Now,calculate the composition $(fog)(x) = f(g(x)) = f(cx + d) = (cx + d) + 2 = cx + d + 2$. Next,calculate the composition $(gof)(x) = g(f(x)) = g(x + 2) = c(x + 2) + d = cx + 2c + d$. Since $(fog)(x) = (gof)(x)$ for all $x$,we equate the two expressions: $cx + d + 2 = cx + 2c + d$. Subtracting $cx$ and $d$ from both sides,we get $2 = 2c$,which implies $c = 1$. Since $d$ cancels out from both sides,$d$ can be any real number (arbitrary). Thus,$c = 1$ and $d$ is arbitrary.

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 $d$?

Similar Questions

Let $f: \{2, 3, 4, 5\} \rightarrow \{3, 4, 5, 9\}$ and $g: \{3, 4, 5, 9\} \rightarrow \{7, 11, 15\}$ be functions defined as $f(2)=3, f(3)=4, f(4)=f(5)=5$ and $g(3)=g(4)=7$ and $g(5)=g(9)=11$. Find $g \circ f$.

If $f: A \rightarrow B$ and $g: B \rightarrow C$ are two functions such that $g \circ f: A \rightarrow C$ is a bijection,then which one of the following is always true?

Let $f(x) = 1 - x$, $g(x) = \frac{1}{1 - x}$, and $h(x) = \frac{1}{x}$ be three functions, for $x \neq 0, 1$. If a function $F(x)$ satisfies $f(F(h(x))) = g(x)$, then which of the following is true?

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x) = x^{2} - 3x + 4$ and $g(x) = 2x + 1$,then the value of $x$ for which $f(x) = (f \circ g)(x)$ is

If $f(x) = \frac{(\tan 1^{\circ}) x + \log_{e}(123)}{x \log_{e}(1234) - (\tan 1^{\circ})}$,$x > 0$,then the least value of $f(f(x)) + f(f(4/x))$ is $...........$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module