Let `f(x) = x^2 and g(x) = sqrt(x)` then which of the following is correct?

Options

A. gof (4) = 4
B. gof (3)=6
C. gof(-2) = -2
D. gof (2) = 4

Answer

To solve the problem, we need to find the composition of the functions \( f(x) \) and \( g(x) \). The functions are defined as follows:

- \( f(x) = x^2 \)
- \( g(x) = \sqrt{x} \)

We want to evaluate \( g(f(x)) \).

### Step 1: Substitute \( f(x) \) into \( g(x) \)

We start by substituting \( f(x) \) into \( g(x) \):

\[ g(f(x)) = g(x^2) \]

### Step 2: Apply the definition of \( g(x) \)

Now, we apply the function \( g \) to \( x^2 \):

\[ g(x^2) = \sqrt{x^2} \]

### Step 3: Simplify \( \sqrt{x^2} \)

The square root of \( x^2 \) is \( |x| \) (the absolute value of \( x \)). Therefore, we have:

\[ g(f(x)) = |x| \]

### Step 4: Analyze the output

The output \( |x| \) is always non-negative, regardless of whether \( x \) is positive or negative. This means that \( g(f(x)) \) will always yield a non-negative result.

### Conclusion

Now, we can check the options provided in the question to find which one is correct based on our result \( g(f(x)) = |x| \).

**Correct Option:**
- If the options include \( g(f(4)) = |4| = 4 \), then this is correct.
- If there are any options that yield a negative result, they are incorrect.