Assume that \(f: (-2, -1) \to (1, 2)\) is an onto function and for \(i = 1, 2, 3, 4\), define \(g_1(x) = f(x) - 2\), \(g_2(x) = f(-x)\), \(g_3(x) = -f(-x)\) and \(g_4(x) = f(-x - 2)\). What is the correct arrangement of \(g_1, g_2, g_3, g_4\) such that the graph of the \(k^{\text{th}}\) function lies in the \(k^{\text{th}}\) quadrant for \(k = 1, 2, 3, 4\)?
Show Hint
- \(g_4, g_1, g_3, g_2\)
- \(g_3, g_4, g_2, g_1\)
- \(g_1, g_2, g_3, g_4\)
- \(g_2, g_4, g_1, g_3\)
The Correct Option is D
Solution and Explanation
We are given the domain and range of a function \(f(x)\) as \(D_f = (-2, -1)\) and \(R_f = (1, 2)\). This implies that for any point \((x, y)\) on \(f(x)\), \(x\) is negative and \(y\) is positive, placing the graph of \(f(x)\) in the 2nd Quadrant. We need to determine the quadrant for four transformed functions \(g_1, g_2, g_3, g_4\) and arrange them such that the \(1^{\text{st}}\) function in the list is in Quadrant 1, the \(2^{\text{nd}}\) in Quadrant 2, and so on. Step 2: Analyzing Each Transformation:
Let's determine the Domain (x-values) and Range (y-values) for each \(g_i(x)\). 1. \(g_1(x) = f(x) - 2\)
Domain: Same as \(f(x)\), so \(x \in (-2, -1)\) (Negative).
Range: \(y \in (1, 2) - 2 \implies y \in (-1, 0)\) (Negative).
Quadrant: \(x<0, y<0 \implies\) 3rd Quadrant.
2. \(g_2(x) = f(-x)\)
Domain: The argument \(-x\) must be in \((-2, -1)\), so \(x \in (1, 2)\) (Positive).
Range: Outputs are directly from \(f(x)\), so \(y \in (1, 2)\) (Positive).
Quadrant: \(x>0, y>0 \implies\) 1st Quadrant.
3. \(g_3(x) = -f(-x)\)
Domain: Same as \(g_2(x)\), so \(x \in (1, 2)\) (Positive).
Range: Outputs are negated values of \(f(x)\), so \(y \in (-2, -1)\) (Negative).
Quadrant: \(x>0, y<0 \implies\) 4th Quadrant.
4. \(g_4(x) = f(-x - 2)\)
Domain: Argument \(-x - 2 \in (-2, -1)\). \[ -2<-x - 2<-1 \] \[ 0<-x<1 \] \[ -1<x<0 \] So \(x \in (-1, 0)\) (Negative).
Range: Outputs are from \(f(x)\), so \(y \in (1, 2)\) (Positive).
Quadrant: \(x<0, y>0 \implies\) 2nd Quadrant.
Step 3: Matching to Quadrants:
We need the order: Quadrant 1, Quadrant 2, Quadrant 3, Quadrant 4.
\(k=1\) (Q1): \(g_2\)
\(k=2\) (Q2): \(g_4\)
\(k=3\) (Q3): \(g_1\)
\(k=4\) (Q4): \(g_3\)
The arrangement is \(g_2, g_4, g_1, g_3\).
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Questions Asked in SRCC GBO exam
Read the given passage and answer the questions.
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