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Select the correct answer.



Consider these functions:

$\begin{array}{l}

f(x)=-2 x-1 \\

g(x)=-\frac{1}{2} x+\frac{1}{2}

\end{array}$



Which statements, if any, are true about these functions?

I. The function $f(g(x))=x$ for all real $x$.

II. The function $g(f(x))=x$ for all real $x$.

III. Functions $f$ and $g$ are inverse functions.


AI only
BII only
CI, II, and III
DNone of the statements are true.

Verified Answer :

  • Evaluate \(f(g(x)) = -2(-\frac{1}{2}x + \frac{1}{2}) - 1 = x - 2\), showing statement I is false.
  • Evaluate \(g(f(x)) = -\frac{1}{2}(-2x - 1) + \frac{1}{2} = x + 1\), showing statement II is false.
  • Conclude that statement III is false since I and II are false.
  • Final answer: \(\boxed{\text{None of the statements are true.}}\)

Explanation

  1. Analysis of the Given Functions

We are given the functions \(f(x) = -2x - 1\) and \(g(x) = -\frac{1}{2}x + \frac{1}{2}\). Our objective is to determine which of the statements I, II, and III are true by evaluating the compositions \(f(g(x))\) and \(g(f(x))\). Recall that two functions are inverse functions if and only if both compositions \(f(g(x)) = x\) and \(g(f(x)) = x\) hold for all real numbers \(x\).

  1. Evaluating Statement I: \(f(g(x)) = x\)

To check Statement I, we calculate \(f(g(x))\) by substituting \(g(x)\) into \(f(x)\):
\[f(g(x)) = f\left(-\frac{1}{2}x + \frac{1}{2}\right) = -2\left(-\frac{1}{2}x + \frac{1}{2}\right) - 1\]
Multiplying the terms:
\[f(g(x)) = \left(-2 \times -\frac{1}{2}x\right) + \left(-2 \times \frac{1}{2}\right) - 1 = x - 1 - 1 = x - 2\]
Since \(f(g(x)) = x - 2\) and not \(x\), Statement I is false.

  1. Evaluating Statement II: \(g(f(x)) = x\)

To check Statement II, we calculate \(g(f(x))\) by substituting \(f(x)\) into \(g(x)\):
\[g(f(x)) = g(-2x - 1) = -\frac{1}{2}(-2x - 1) + \frac{1}{2}\]
Multiplying the terms:
\[g(f(x)) = \left(-\frac{1}{2} \times -2x\right) + \left(-\frac{1}{2} \times -1\right) + \frac{1}{2} = x + \frac{1}{2} + \frac{1}{2} = x + 1\]
Since \(g(f(x)) = x + 1\) and not \(x\), Statement II is false.

  1. Conclusion on Statement III and Final Result

Statement III claims that \(f\) and \(g\) are inverse functions. By the definition of inverse functions, this would require both \(f(g(x)) = x\) and \(g(f(x)) = x\). Since we found both I and II to be false, Statement III is also false. Consequently, none of the statements are true.

Examples

Inverse functions are essential in real-world scenarios, such as converting between different units of measure. For instance, if a function \(f(x)\) converts temperatures from Celsius to Fahrenheit, its inverse \(g(x)\) must convert Fahrenheit back to Celsius exactly. By testing composite functions \(f(g(x))\) and \(g(f(x))\), you verify that the conversion process is reversible and accurate, which is vital for consistency in scientific and engineering applications.

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