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.
Top Questions on Functions
- The value of \( \frac{100 C_{50}}{51} + \frac{100 C_{51}}{52} + \dots + \frac{100 C_{100}}{101} \) is :
- Let the line \( y - x = 1 \) intersect the ellipse \( \frac{x^2}{2} + \frac{y^2}{1} = 1 \) at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is:
- Let \( f \) be a polynomial function such that \[ f(x^2+1)=x^4+5x^2+2,\quad \text{for all } x\in\mathbb{R}. \] Then \[ \int_0^3 f(x)\,dx \] is equal to:
- If \( g(x) = 3x^2 + 2x - 3 \), \( f(0) = -3 \) and \( 4g(f(x)) = 3x^2 - 32x + 72 \), then \( f(g(2)) \) is equal to:
If the domain of the function \[ f(x)=\log\left(10x^2-17x+7\right)\left(18x^2-11x+1\right) \] is $(-\infty,a)\cup(b,c)\cup(d,\infty)-\{e\}$, then $90(a+b+c+d+e)$ equals
Questions Asked in CAT exam
- The passage given below is followed by four summaries. Choose the option that best captures the essence of the passage.
In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
Cultural appropriation occurs when elements from a particular culture are borrowed without proper understanding, respect, or acknowledgment. This leads to the commodification of sacred symbols, the reinforcement of stereotypes, and the erasure of the cultural context from which these elements originated. It is essential to recognize that the impact of cultural appropriation extends beyond the realm of artistic expression, influencing societal perceptions and perpetuating power imbalances.- CAT - 2025
- Para Summary
- The number of distinct integers $n$ for which $\log_{\left(\frac14\right)}(n^2 - 7n + 11)>0$ is:
- CAT - 2025
- Linear Inequalities
- In the set of consecutive odd numbers $\{1, 3, 5, \ldots, 57\}$, there is a number $k$ such that the sum of all the elements less than $k$ is equal to the sum of all the elements greater than $k$. Then, $k$ equals?
- CAT - 2025
- Number Systems
- The number of distinct pairs of integers $(x, y)$ satisfying the inequalities $x>y \ge 3$ and $x + y<14$ is:
- CAT - 2025
- Number Systems
For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is:
- CAT - 2025
- Linear Inequalities