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]
