Question:
If \( f(x) = |\cos x| \) and \( g(x) = [x] \), then \( g \circ f(x) \) is equal to
If \( f(x) = |\cos x| \) and \( g(x) = [x] \), then \( g \circ f(x) \) is equal to
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When working with the floor function and absolute value, remember that the floor function of a value between 0 and 1 is always 0.
Updated On: Apr 22, 2026
- \( |\cos [x]| \)
- \( \cos [x] \)
- \( [\cos x] \)
- \( [|\cos x|] \)
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The Correct Option is C
Solution and Explanation
Step 1: Define the composition of functions.
We are given two functions:
- \( f(x) = |\cos x| \), which gives the absolute value of the cosine of \( x \),
- \( g(x) = [x] \), which is the greatest integer less than or equal to \( x \), i.e., the floor function.
We are asked to find \( g \circ f(x) \), which is defined as \( g(f(x)) \), i.e., \( g(\cos x) \) for \( f(x) = |\cos x| \).
Step 2: Apply \( f(x) \).
First, we apply \( f(x) \): \[ f(x) = |\cos x| \] Thus, \( f(x) \) gives the absolute value of \( \cos x \), which is always non-negative.
Step 3: Apply \( g(x) \).
Next, apply \( g(x) = [x] \) to \( f(x) = |\cos x| \). We have: \[ g(f(x)) = g(|\cos x|) = [|\cos x|] \] Since \( |\cos x| \) is always between 0 and 1 for all values of \( x \), the greatest integer less than or equal to \( |\cos x| \) is 0 for all \( x \) where \( |\cos x| \neq 1 \).
Step 4: Conclusion.
Thus, \( g(f(x)) = [|\cos x|] \), which is equivalent to option (C), \( [\cos x] \).
We are given two functions:
- \( f(x) = |\cos x| \), which gives the absolute value of the cosine of \( x \),
- \( g(x) = [x] \), which is the greatest integer less than or equal to \( x \), i.e., the floor function.
We are asked to find \( g \circ f(x) \), which is defined as \( g(f(x)) \), i.e., \( g(\cos x) \) for \( f(x) = |\cos x| \).
Step 2: Apply \( f(x) \).
First, we apply \( f(x) \): \[ f(x) = |\cos x| \] Thus, \( f(x) \) gives the absolute value of \( \cos x \), which is always non-negative.
Step 3: Apply \( g(x) \).
Next, apply \( g(x) = [x] \) to \( f(x) = |\cos x| \). We have: \[ g(f(x)) = g(|\cos x|) = [|\cos x|] \] Since \( |\cos x| \) is always between 0 and 1 for all values of \( x \), the greatest integer less than or equal to \( |\cos x| \) is 0 for all \( x \) where \( |\cos x| \neq 1 \).
Step 4: Conclusion.
Thus, \( g(f(x)) = [|\cos x|] \), which is equivalent to option (C), \( [\cos x] \).
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