Select the correct answer.

Which equation written in two variables is equivalent to this function?

[tex]\[ f(x) = -2(x - 4) \][/tex]

A. [tex]\[ y = -2(f(x) - 4) \][/tex]

B. [tex]\[ y = -2x + 4 \][/tex]

C. [tex]\[ y = -2(f(x) + 4) \][/tex]

D. [tex]\[ y = -2(x - 4) \][/tex]



Answer :

Let's determine the equation in two variables that is equivalent to the given function [tex]\( f(x) = -2(x - 4) \)[/tex].

### Step-by-Step Rationalization:

1. Given Function:
[tex]\[ f(x) = -2(x - 4) \][/tex]

2. Rewrite the Function in Terms of [tex]\( y \)[/tex]:
Let's rewrite the function [tex]\( f(x) \)[/tex] as [tex]\( y \)[/tex]:
[tex]\[ y = -2(x - 4) \][/tex]

3. Examine the Options:
Now, let's examine the given options and see which one matches the equation [tex]\( y = -2(x - 4) \)[/tex]:

Option 1: [tex]\( y = -2(f(x) - 4) \)[/tex]
- This option implies, substituting [tex]\( f(x) \)[/tex]:
[tex]\[ y = -2(-2(x - 4) - 4) \][/tex]
- Simplify the expression inside the parentheses:
[tex]\[ y = -2(-2x + 8 - 4) \][/tex]
[tex]\[ y = -2(-2x + 4) \][/tex]
[tex]\[ y = 4x - 8 \][/tex]
This doesn't match [tex]\( y = -2(x - 4) \)[/tex].

Option 2: [tex]\( y = -2x + 4 \)[/tex]
- This equation simplifies to:
[tex]\[ y = -2x + 4 \][/tex]
This doesn't match [tex]\( y = -2(x - 4) \)[/tex].

Option 3: [tex]\( y = -2(f(x) + 4) \)[/tex]
- This option implies, substituting [tex]\( f(x) \)[/tex]:
[tex]\[ y = -2(-2(x - 4) + 4) \][/tex]
- Simplify the expression inside the parentheses:
[tex]\[ y = -2(-2x + 8 + 4) \][/tex]
[tex]\[ y = -2(-2x + 12) \][/tex]
[tex]\[ y = 4x - 24 \][/tex]
This doesn't match [tex]\( y = -2(x - 4) \)[/tex].

Option 4: [tex]\( y = -2(x - 4) \)[/tex]
- This option matches directly with the given function rewritten with [tex]\( y \)[/tex]:
[tex]\[ y = -2(x - 4) \][/tex]

4. Conclusion:
Therefore, the correct equation in two variables that is equivalent to [tex]\( f(x) = -2(x - 4) \)[/tex] is:
[tex]\[ y = -2(x - 4) \][/tex]

Hence, the correct answer is the fourth option:
[tex]\[ y = -2(x - 4) \][/tex]