select the correct answer.\nconsider these functions:\n$f(x) = -2x - 1$\n$g(x) = -\frac{1}{2}x +…

select the correct answer.\nconsider these functions:\n$f(x) = -2x - 1$\n$g(x) = -\frac{1}{2}x + \frac{1}{2}$\nwhich statements, if any, are true about these functions?\ni. the function $f(g(x)) = x$ for all real $x$.\nii. the function $g(f(x)) = x$ for all real $x$.\niii. functions $f$ and $g$ are inverse functions.\na. i only\nb. ii only\nc. i, ii, and iii\nd. none of the statements are true.

select the correct answer.\nconsider these functions:\n$f(x) = -2x - 1$\n$g(x) = -\frac{1}{2}x + \frac{1}{2}$\nwhich statements, if any, are true about these functions?\ni. the function $f(g(x)) = x$ for all real $x$.\nii. the function $g(f(x)) = x$ for all real $x$.\niii. functions $f$ and $g$ are inverse functions.\na. i only\nb. ii only\nc. i, ii, and iii\nd. none of the statements are true.

Answer

Explanation:

Step1: Calculate ( f(g(x)) )

Substitute ( g(x) ) into ( f(x) ):
( f(g(x)) = f\left(-\frac{1}{2}x + \frac{1}{2}\right) = -2\left(-\frac{1}{2}x + \frac{1}{2}\right) - 1 )
Simplify: ( -2\left(-\frac{1}{2}x\right) - 2\left(\frac{1}{2}\right) - 1 = x - 1 - 1 = x - 2 \neq x ). So Statement I is false.

Step2: Calculate ( g(f(x)) )

Substitute ( f(x) ) into ( g(x) ):
( g(f(x)) = g(-2x - 1) = -\frac{1}{2}(-2x - 1) + \frac{1}{2} )
Simplify: ( x + \frac{1}{2} + \frac{1}{2} = x + 1 \neq x ). So Statement II is false.

Step3: Check Inverse Function Definition

For ( f ) and ( g ) to be inverses, ( f(g(x)) = x ) and ( g(f(x)) = x ) must both hold. Since neither holds, Statement III is false.

Answer:

D. None of the statements are true.