Question

what is $f(x) + f(x) + f(x)$?
complete
$3f(x) = $
done
$3x$
$3x^2$
$x^2 + 3$

Explanation:

To determine \( 3f(x) \), we first need to know the definition of \( f(x) \). However, since the problem is likely assuming a general form or perhaps a common function (though not fully specified here), but looking at the options, we can analyze:

If \( f(x) = x \), then \( 3f(x) = 3x \). If \( f(x) = x^2 \), then \( 3f(x) = 3x^2 \), and if \( f(x)=\frac{x^2 + 3}{3}\) (not a standard case), but from the options, we assume that based on the context (maybe \( f(x)=x \) or \( f(x)=x^2 \) or other). But since the first part \( f(x)+f(x)+f(x) = 3f(x) \) is established, and the options are \( 3x \), \( 3x^2 \), \( x^2 + 3 \).

If we assume \( f(x)=x \), then \( 3f(x) = 3x \). If \( f(x)=x^2 \), then \( 3f(x)=3x^2 \). But since the problem is likely a basic function, and the first step is combining like terms (summing \( f(x) \) three times gives \( 3f(x) \)), and then depending on \( f(x) \)'s form.

But since the options are given, and assuming that maybe \( f(x) = x \) (so \( 3f(x)=3x \)) or \( f(x)=x^2 \) (so \( 3f(x)=3x^2 \)) or \( f(x)=\frac{x^2 + 3}{3} \) (but that's less likely). However, since the first part is completed as \( 3f(x) \), and the options are there, we need to match.

Wait, maybe the original \( f(x) \) was, for example, if \( f(x) = x \), then \( 3f(x) = 3x \); if \( f(x) = x^2 \), then \( 3f(x) = 3x^2 \); if \( f(x) = \frac{x^2 + 3}{3} \), then \( 3f(x) = x^2 + 3 \), but that's a stretch.

But since the problem is presented as a follow - up to \( f(x)+f(x)+f(x)=3f(x) \), and the options are given, we can consider that if we take \( f(x)=x \), then \( 3f(x)=3x \); if \( f(x)=x^2 \), then \( 3f(x)=3x^2 \); and \( x^2 + 3 \) would be \( 3f(x) \) only if \( f(x)=\frac{x^2 + 3}{3} \), which is not a typical simple function.

Assuming a simple linear or quadratic function, and since the first step is combining \( f(x) \) three times, the most probable answers (depending on \( f(x) \)):

If \( f(x)=x \), then \( 3f(x)=3x \)

If \( f(x)=x^2 \), then \( 3f(x)=3x^2 \)

If we assume \( f(x) \) is linear (like \( f(x)=x \)), then \( 3f(x)=3x \); if \( f(x) \) is quadratic (like \( f(x)=x^2 \)), then \( 3f(x)=3x^2 \)

Case 1: If \( f(x)=x \)

Step 1: Define \( f(x) \)

Let \( f(x)=x \)

Step 2: Calculate \( 3f(x) \)

Using the formula for scalar multiplication of a function, \( 3f(x)=3\times f(x) \). Substituting \( f(x) = x \), we get \( 3f(x)=3\times x=3x \)

Case 2: If \( f(x)=x^2 \)

Step 1: Define \( f(x) \)

Let \( f(x)=x^2 \)

Step 2: Calculate \( 3f(x) \)

Using the formula for scalar multiplication of a function, \( 3f(x)=3\times f(x) \). Substituting \( f(x)=x^2 \), we get \( 3f(x)=3\times x^2 = 3x^2 \)

Case 3: If \( f(x)=\frac{x^2 + 3}{3} \)

Step 1: Define \( f(x) \)

Let \( f(x)=\frac{x^2+3}{3} \)

Step 2: Calculate \( 3f(x) \)

Using the formula for scalar multiplication of a function, \( 3f(x)=3\times f(x) \). Substituting \( f(x)=\frac{x^2 + 3}{3} \), we get \( 3f(x)=3\times\frac{x^2 + 3}{3}=x^2 + 3 \)

Since the problem is likely a basic function (linear or quadratic) and the first step is combining \( f(x) \) three times, the possible answers are:

If we consider a linear function \( f(x)=x \), the answer is \( 3x \); if we consider a quadratic function \( f(x)=x^2 \), the answer is \( 3x^2 \); and \( x^2 + 3 \) is also an option.

But since the problem is presented in a basic math context (probably for a beginner), and the first step is combining \( f(x) \) three times, the most probable answers (depending on \( f(x) \)):

If we assume \( f(x)=x \), then \( 3f(x)=\boldsymbol{3x} \)

If we assume \( f(x)=x^2 \), then \( 3f(x)=\boldsymbol{3x^2} \)

If we assume \( f(x)=\frac{x^2 + 3}{3} \), then \( 3f(x)=\boldsymbol{x^2 + 3} \)

However, since the problem is a follow - up to \( f(x)+f(x)+f(x)=3f(x) \), and considering the options, if we take a simple function like \( f(x)=x \) (a common basic function), then \( 3f(x)=3x \); if \( f(x)=x^2 \), then \( 3f(x)=3x^2 \).

Answer:

Step 1: Define \( f(x) \)

Let \( f(x)=\frac{x^2+3}{3} \)

Step 2: Calculate \( 3f(x) \)

Using the formula for scalar multiplication of a function, \( 3f(x)=3\times f(x) \). Substituting \( f(x)=\frac{x^2 + 3}{3} \), we get \( 3f(x)=3\times\frac{x^2 + 3}{3}=x^2 + 3 \)

Since the problem is likely a basic function (linear or quadratic) and the first step is combining \( f(x) \) three times, the possible answers are:

If we consider a linear function \( f(x)=x \), the answer is \( 3x \); if we consider a quadratic function \( f(x)=x^2 \), the answer is \( 3x^2 \); and \( x^2 + 3 \) is also an option.

But since the problem is presented in a basic math context (probably for a beginner), and the first step is combining \( f(x) \) three times, the most probable answers (depending on \( f(x) \)):

If we assume \( f(x)=x \), then \( 3f(x)=\boldsymbol{3x} \)

If we assume \( f(x)=x^2 \), then \( 3f(x)=\boldsymbol{3x^2} \)

If we assume \( f(x)=\frac{x^2 + 3}{3} \), then \( 3f(x)=\boldsymbol{x^2 + 3} \)

However, since the problem is a follow - up to \( f(x)+f(x)+f(x)=3f(x) \), and considering the options, if we take a simple function like \( f(x)=x \) (a common basic function), then \( 3f(x)=3x \); if \( f(x)=x^2 \), then \( 3f(x)=3x^2 \).