Which one of the given figures P, Q, R, and S represents the graph of the following function?
\[
f(x) = |\,|x+2| - |x-1|\,|
\]
\begin{center}
\includegraphics[width=0.5\textwidth]{02.jpeg}
\end{center}
Show Hint
- P
- Q
- R
- S
The Correct Option is C
Solution and Explanation
Step 1: Identify critical points. Absolute value expressions change at \(x = -2\) and \(x = 1\). So we break into intervals: \(-\infty < x < -2\), \(-2 \leq x < 1\), and \(x \geq 1\).
Step 2: Case 1: \(x \leq -2\). \(|x+2| = -(x+2), \quad |x-1| = -(x-1)\). So, \[ f(x) = |[-(x+2)] - [-(x-1)]| = | -x-2 + x -1 | = |-3| = 3 \] Thus, \(f(x) = 3\) (constant).
Step 3: Case 2: \(-2 \leq x < 1\). \(|x+2| = x+2, \quad |x-1| = -(x-1)\). So, \[ f(x) = |(x+2) - (-(x-1))| = |x+2 + x -1| = |2x+1| \] For \(-2 \leq x < 1\), \(2x+1\) varies from \(-3\) to \(3\). Thus, \(f(x) = |2x+1|\), V-shaped graph passing through \((-0.5, 0)\).
Step 4: Case 3: \(x \geq 1\). \(|x+2| = x+2, \quad |x-1| = x-1\). So, \[ f(x) = |(x+2) - (x-1)| = |3| = 3 \] Thus, \(f(x) = 3\) (constant).
Step 5: Combine. - For \(x \leq -2\): flat line at 3. - For \(-2 \leq x < 1\): V-shaped \(|2x+1|\). - For \(x \geq 1\): flat line at 3. This matches the shape of **Graph R** in the options.
Final Answer: \[ \boxed{R} \]
Top Questions on Geometry
- The equation of chord AB of ellipse \(2x^2 + y^2 = 1\) is \(x - y + 1 = 0\). If O is the origin, then \(\angle AOB =\)
In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of \( P \), \( Q \), and \( R \)?
- Let the sum of the focal distances of the point $ P(4, 3) $ on the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be $ 8\sqrt{\frac{5}{3}} $. If for $ H $, the length of the latus rectum is $ \ell $ and the product of the focal distances of the point $ P $ is $ m $, then $ 9\ell^2 + 6m $ is equal to:
- Let for two distinct values of $ p $, the lines $ y = x + p $ touch the ellipse $ E: \frac{x^2}{4} + \frac{y^2}{9} = 1 $ at the points $ A $ and $ B $. Let the line $ y = x $ intersect $ E $ at the points $ C $ and $ D $. Then the area of the quadrilateral $ ABCD $ is equal to:
The center of a circle $ C $ is at the center of the ellipse $ E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $, where $ a>b $. Let $ C $ pass through the foci $ F_1 $ and $ F_2 $ of $ E $ such that the circle $ C $ and the ellipse $ E $ intersect at four points. Let $ P $ be one of these four points. If the area of the triangle $ PF_1F_2 $ is 30 and the length of the major axis of $ E $ is 17, then the distance between the foci of $ E $ is:
Questions Asked in GATE GG exam
The figures I, II, and III are parts of a sequence. Which one of the following options comes next in the sequence at IV?
- GATE CH - 2025
- GATE BT - 2025
- GATE GG - 2025
- Sequence and Series
A color model is shown in the figure with color codes: Yellow (Y), Magenta (M), Cyan (Cy), Red (R), Blue (Bl), Green (G), and Black (K). Which one of the following options displays the color codes that are consistent with the color model?
- A thin wire is used to construct all the edges of a cube of 1 m side by bending, cutting, and soldering the wire. If the wire is 12 m long, what is the minimum number of cuts required to construct the wire frame to form the cube?
- Rohit goes to a restaurant for lunch at about 1 PM. When he enters the restaurant, he notices that the hour and minute hands on the wall clock are exactly coinciding. After about an hour, when he leaves the restaurant, he notices that the clock hands are again exactly coinciding. How much time (in minutes) did Rohit spend at the restaurant?
Is there any good show __________ television tonight? Select the most appropriate option to complete the above sentence.
- GATE CH - 2025
- GATE BT - 2025
- GATE GG - 2025
- English Grammar