Find $g$ of and $f \circ g$, if $f: \mathbf{R} \rightarrow \mathbf{R}$ and $g: \mathbf{R} \rightarrow \mathbf{R}$ are given by $f(x)=\cos x$ and $g(x)=3 x^{2}$. Show that $g$ of $\neq$ fog.

Step 1: Define the functions

We are given two functions, $f(x) = \cos x$ and $g(x) = 3x^2$. Both functions map real numbers to real numbers.

Step 2: Calculate $g \circ f$

To find $g \circ f(x)$, we substitute $f(x)$ into $g(x)$. Since $f(x) = \cos x$, we replace $x$ in $g(x)$ with $\cos x$. This gives us $3(\cos x)^2$, which simplifies to $3\cos^2 x$.

Step 3: Calculate $f \circ g$

To find $f \circ g(x)$, we substitute $g(x)$ into $f(x)$. Since $g(x) = 3x^2$, we replace $x$ in $f(x)$ with $3x^2$. This results in $\cos(3x^2)$.

Step 4: Compare $g \circ f$ and $f \circ g$

By comparing the results, we see that $g \circ f(x) = 3\cos^2 x$ and $f \circ g(x) = \cos(3x^2)$. These two expressions are generally not equal for all values of $x$. For example, if $x=0$, $g \circ f(0) = 3\cos^2(0) = 3(1)^2 = 3$, while $f \circ g(0) = \cos(3(0)^2) = \cos(0) = 1$. Since $3 \neq 1$, we have shown that $g \circ f \neq f \circ g$.

More from this chapter

• Show that an onto function f:1,2,3 →1,2,3 is always one-one.
• Let A=-2,-1,0,1,2,3. Define a relation R on A by xRy if and only if y=max(x,1). If l is the number of elements in R, and m and n are the minimum numbers of orde
• 32. Show that f: R -> R defined as f(x) = x / sqrt(1 + x²) is one-one but not onto. (function, one-one, onto, real numbers, domain, codomain, range)
• Let A be the set of all students of a boys school. Show that the relation R in A given by R=(a, b): a is sister of b is the empty relation and R^()=(a, b): the
• Consider the identity function I_N: N → N defined as I_N(x)=x x N. Show that although I_N is onto but I_N+I_N: N → N defined as (I_N+I_N)(x)=I_N(x)+I_N(x)=x+x=2
• Prove that the relation R in the set A = 1, 2, 3, 4, 5 given by R = (a, b): |a - b| is even is an equivalence relation.
• Let f(x) = log_e x and g(x) = (x^4 - 2x³ + 3x² - 2x + 2)/(2x² - 2x + 1). Then the domain of f ° g is
• Let the domain of the function f(x) = cos^(-1) ( (4x + 5)/(3x - 7)) be [α, β] and the domain of g(x) = log_2(2 - 6log_7(2x + 5)) be [, ]. Then |7(α + β) + 4( +)

All Relations and Functions solutions →