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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- Functions \( g \) and \( h \) are defined on \( n \) constants, \( a_0, a_1, a_2, a_3, \ldots, a_{n-1} \), as follows:
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Let \( f \) be an injective map with domain x, y, z and range 1, 2, 3 such that exactly one of the following statements is correct and the remaining are false.}
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