If f:R→R, g(x)=3x2+7 and f(x)=√x, then gοf(x) is equal to

The correct option is 3x+7.

To find the composition of the functions, we need to understand the definition of composite functions. Let's break down the solution step-by-step.

Step 1: Identify the given functions
We are given two functions:

$$f\left(x\right)=\sqrt{x}$$

and

$$g\left(x\right)=3x^2+7$$

Step 2: Understand the composite function definition
The composition of function g with f, denoted by $(g\circ f)\left(x\right)$, is defined as:

$$(g\circ f)\left(x\right)=g\left(f\right(x\left)\right)$$

This means we substitute the entire function $f\left(x\right)$ as the input argument into the function $g\left(x\right)$.

Step 3: Substitute the value of f(x) into g(x)
By substituting $f\left(x\right)=\sqrt{x}$ into the expression, we get:

$$g\left(f\right(x\left)\right)=g\left(\sqrt{x}\right)$$

Now, we evaluate $g\left(\sqrt{x}\right)$ by replacing every occurrence of $x$ in the formula for $g\left(x\right)$ with $\sqrt{x}$:

$$g\left(\sqrt{x}\right)=3{\left(\sqrt{x}\right)}^2+7$$

Step 4: Simplify the expression
We simplify the square of the square root of x. For all non-negative real numbers x:

$${\left(\sqrt{x}\right)}^2=x$$

Substituting this back into our composition equation gives:

$$g\left(f\right(x\left)\right)=3x+7$$

Thus, $(g\circ f)\left(x\right)$ is equal to $3x+7$.