Let \(f(x)\)={-\(5,x≤0 x-5,x>0\) and \(g(x)=If(x)+2f(IxI)\).Then \(g(-2)\) will be
\(-15\)
\(1\)
\(0\)
\(-11\)
\(-1\)
The Correct Option is
Approach Solution - 1
Given the functions:
\[ f(x) = \begin{cases} -5, & x \leq 0 \\ x - 5, & x > 0 \end{cases} \] \[ g(x) = |f(x)| + 2f(|x|) \]
We want to find \( g(-2) \). Let's break it down:
- \( f(-2) \): Since \( -2 \leq 0 \), \( f(-2) = -5 \).
- \( |f(-2)| \): \( |-5| = 5 \).
- \( |x| \) when \( x = -2 \): \( |-2| = 2 \).
- \( f(|x|) = f(2) \):
- Since \( 2 > 0 \), \( f(2) = 2 - 5 = -3 \).
- \( 2f(|x|) = 2f(2) \): \( 2 \cdot (-3) = -6 \).
- \( g(-2) = |f(-2)| + 2f(|-2|) \): \( g(-2) = 5 + (-6) = -1 \).
Therefore, \( g(-2) = -1 \).
Approach Solution -2
We can approach this problem by directly substituting \( x = -2 \) into the expression for \( g(x) \) and using the definition of \( f(x) \) piecewise.
\[ g(x) = |f(x)| + 2f(|x|) \] \[ g(-2) = |f(-2)| + 2f(|-2|) \]
Since \( -2 \leq 0 \), \( f(-2) = -5 \). Therefore, \( |f(-2)| = |-5| = 5 \).
Since \( |-2| = 2 \), we need to evaluate \( f(2) \). Since \( 2 > 0 \), \( f(2) = 2 - 5 = -3 \).
Substituting back into the expression for \( g(-2) \):
\[ g(-2) = 5 + 2(-3) = 5 - 6 = -1 \]
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Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
