Graphs of functions are given. Mark option
Graphs of functions are given. Mark option
Show Hint
- If f(x) = 3f(−x)
- If f(x) = f(−x)
- If f(x) = −f(−x)
- If 3f(x) = 6f(−x)
The Correct Option is A
Solution and Explanation
From the graph we observe:
- For \( x > 0 \), the function increases linearly with slope \( +1 \) starting from the origin. Therefore: \[ f(1) = 1 \]
- For \( x < 0 \), the function decreases with slope \( -2 \), also starting from the origin. Therefore: \[ f(-1) = 2 \]
Step 1: Compare Values at \( x = 1 \) and \( x = -1 \)
Using the values: \[ f(1) = 1, \quad f(-1) = 2 \] We find a relationship: \[ f(-1) = 2f(1) \quad \Rightarrow \quad f(1) = \frac{1}{2} f(-1) \]
Step 2: Evaluate the Given Options
Suppose the function satisfies \( f(x) = 3f(-x) \). Then: \[ f(1) = 3f(-1) \quad \Rightarrow \quad 1 = 3 \times 2 = 6 \quad \text{(Incorrect)} \] Now try: \[ f(x) = \frac{1}{2} f(-x) \quad \Rightarrow \quad f(1) = \frac{1}{2} \times f(-1) = \frac{1}{2} \times 2 = 1 \quad \text{(Correct)} \] So this relation works. That means: \[ f(-x) = 2f(x) \] Which is equivalent to: \[ \boxed{f(x) = \frac{1}{2}f(-x)} \] So the best matching option among the choices is the one where: \[ f(x) = \frac{1}{2} f(-x) \] or equivalently: \[ f(-x) = 2 f(x) \]
Conclusion
Therefore, the correct functional relationship supported by the graph is: \[ \boxed{f(x) = \frac{1}{2} f(-x)} \quad \text{or} \quad \boxed{f(-x) = 2f(x)} \] Among multiple-choice options, this matches the option where the **negative side is scaled by 2** relative to the positive.
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Questions Asked in CAT exam
The passage below is accompanied by four questions. Based on the passage, choose the best answer for each question.
In the summer of 2022, subscribers to the US streaming service HBO MAX were alarmed to discover that dozens of the platform’s offerings – from the Covid-themed heist thriller Locked Down to the recent remake of The Witches – had been quietly removed from the service . . . The news seemed like vindication to those who had long warned that streaming was more about controlling access to the cultural commons than expanding it, as did reports (since denied by the show’s creators) that Netflix had begun editing old episodes of Stranger Things to retroactively improve their visual effects.
What’s less clear is whether the commonly prescribed cure for these cultural ills – a return to the material pleasures of physical media – is the right one. While the makers of Blu-ray discs claim they have a shelf life of 100 years, such statistics remain largely theoretical until they come to pass, and are dependent on storage conditions, not to mention the continued availability of playback equipment. The humble DVD has already proved far less resilient, with many early releases already beginning to deteriorate in quality Digital movie purchases provide even less security. Any film “bought” on iTunes could disappear if you move to another territory with a different rights agreement and try to redownload it. It’s a bold new frontier in the commodification of art: the birth of the product recall. After a man took to Twitter to bemoan losing access to Cars 2 after moving from Canada to Australia, Apple clarified that users who downloaded films to their devices would retain permanent access to those downloads, even if they relocated to a hemisphere where the [content was] subject to a different set of rights agreements. Thanks to the company’s ironclad digital rights management technology, however, such files cannot be moved or backed up, locking you into watching with your Apple account.
Anyone who does manage to acquire Digital Rights Management free (DRM-free) copies of their favourite films must nonetheless grapple with ever-changing file format standards, not to mention data decay – the gradual process by which electronic information slowly but surely corrupts. Only the regular migration of files from hard drive to hard drive can delay the inevitable, in a sisyphean battle against the ravages of digital time.
In a sense, none of this is new. Charlie Chaplin burned the negative of his 1926 film A Woman of the Sea as a tax write-off. Many more films have been lost through accident, negligence or plain indifference. During a heatwave in July 1937, a Fox film vault in New Jersey burned down, destroying a majority of the silent films produced by the studio.
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