The composite mapping fog, of the maps f: `R to R`, f (x)=sin x, g : `R to R`, g(x) = `x^2 `, is

Options

A. `sin x +x^2`
B. ` (sin x)^2`
C. `sin x^2`
D. `(sin x )/(x^2)`

Answer

To find the composite mapping \( f \circ g \) of the functions \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \sin x \) and \( g: \mathbb{R} \to \mathbb{R} \) defined by \( g(x) = x^2 \), we will follow these steps:

### Step 1: Understand the Composite Function
The composite function \( f \circ g \) means that we will apply the function \( g \) first and then apply the function \( f \) to the result of \( g \). Mathematically, this is represented as:
\[ f \circ g (x) = f(g(x)) \]

### Step 2: Substitute \( g(x) \) into \( f \)
We know that \( g(x) = x^2 \). Now we will substitute \( g(x) \) into \( f \):
\[ f(g(x)) = f(x^2) \]

### Step 3: Apply the Function \( f \)
Now, we will apply the function \( f \) to \( x^2 \). Since \( f(x) = \sin x \), we have:
\[ f(x^2) = \sin(x^2) \]

### Step 4: Write the Final Result
Thus, the composite mapping \( f \circ g \) is:
\[ f \circ g (x) = \sin(x^2) \]

### Conclusion
The composite mapping \( f \circ g \) is given by:
\[ f \circ g (x) = \sin(x^2) \]

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