Understanding the Problem
We are given two linear functions, \(f(x) = ax-b\) and \(g(x) = cx + d\). The problem states that these functions commute under composition, meaning \(f(g(x)) = g(f(x))\). Our goal is to find which of the given mathematical statements is true as a consequence of this condition.
Deriving the Condition for Commutativity
To find the relationship between the coefficients \(a, b, c, d\), we first compute the compositions \(f(g(x))\) and \(g(f(x))\):
- Calculate \(f(g(x))\): Substitute \(g(x)\) into \(f(x)\): \(f(g(x)) = f(cx + d)\) \(f(cx + d) = a(cx + d) - b\) \(f(g(x)) = acx + ad - b\)
- Calculate \(g(f(x))\): Substitute \(f(x)\) into \(g(x)\): \(g(f(x)) = g(ax - b)\) \(g(ax - b) = c(ax - b) + d\) \(g(f(x)) = acx - cb + d\)
- Equate the compositions: Since \(f(g(x)) = g(f(x))\), we set the results equal: \(acx + ad - b = acx - cb + d\)
- Simplify the equation: Subtract \(acx\) from both sides: \(ad - b = -cb + d\) Rearrange the terms to find the core relationship: \(ad + cb = d + b\) This equation, \(ad + cb = d + b\), must hold true for the functions \(f(x)\) and \(g(x)\) to commute.
Evaluating the Options
Now, let's use the derived relationship \(ad + cb = d + b\) to check each option:
Option 1: \(f(d) = g(b)\)
Calculate \(f(d)\) and \(g(b)\):
- \(f(d) = a(d) - b = ad - b\)
- \(g(b) = c(b) + d = cb + d\)
The statement becomes \(ad - b = cb + d\). Rearranging gives \(ad - cb = d + b\). This is not the same as our derived condition \(ad + cb = d + b\). Therefore, this option is incorrect.
Option 2: \(f(b) + g(d) = 0\)
Calculate \(f(b)\) and \(g(d)\):
- \(f(b) = a(b) - b = ab - b\)
- \(g(d) = c(d) + d = cd + d\)
The statement becomes \((ab - b) + (cd + d) = 0\). This expression does not simplify to match our condition \(ad + cb = d + b\). Therefore, this option is incorrect.
Option 3: \(f(a) + g(c) = 2a\)
Calculate \(f(a)\) and \(g(c)\):
- \(f(a) = a(a) - b = a^2 - b\)
- \(g(c) = c(c) + d = c^2 + d\)
The statement becomes \((a^2 - b) + (c^2 + d) = 2a\). This expression does not simplify to match our condition \(ad + cb = d + b\). Therefore, this option is incorrect.
Option 4: \(f(d) + g(b) = 2d\)
Calculate \(f(d)\) and \(g(b)\):
- \(f(d) = a(d) - b = ad - b\)
- \(g(b) = c(b) + d = cb + d\)
Now, let's find the sum \(f(d) + g(b)\):
\(f(d) + g(b) = (ad - b) + (cb + d)\) \(f(d) + g(b) = ad + cb - b + d\)We know from our derivation that \(ad + cb = d + b\). Substitute this into the equation:
\(f(d) + g(b) = (d + b) - b + d\) \(f(d) + g(b) = d + d\) \(f(d) + g(b) = 2d\)This result exactly matches the statement in Option 4. Therefore, this option holds true.
Conclusion
The condition \(f(g(x)) = g(f(x))\) for the given linear functions \(f(x) = ax-b\) and \(g(x) = cx+d\) implies the relationship \(ad + cb = d + b\). By substituting the function definitions into the options and using this relationship, we found that the statement \(f(d) + g(b) = 2d\) is the one that holds true.
Similar Questions
- Let P and Q be two non-void relations on a set A. Which of the following statements are correct?
I. P and Q are reflexive \(\Rightarrow P \cap Q\) is reflexive.
II. P and Q are symmetric \(\Rightarrow P \cup Q\) is symmetric.
III. P and Q are transitive \(\Rightarrow P \cap Q\) is transitive.
Select the answer using the code given below. - Let \(A = \{1, 2, 3, 4, 5\}\) and \(B = \{6, 7\}\). What is the number of onto functions from \(A\) to \(B\)?
- Let \(A = \{x \in R: -1 > x < 1\}\). Which of the following is/are bijective functions from \(A\) to itself?
1. \(f(x) = x|x|\)
2. \(g(x) = \cos(\pi x)\)
Select the correct answer using the code given below : - Let \(R\) be a relation on the open interval \((-1, 1)\) and is given by \(R = \{(x, y) : |x + y| < 2\}\). Then which one of the following is correct?
- Let R be a relation on the set N of natural numbers defined by
\(R = \{(x, y) : x, y \in N \text{ and } x = y^3\}\).
Which of the following statements is/are correct ?
I. R is symmetric relation
II. R is transitive relation
Select the answer using the code given below : - Let \(A = \{-3,-2,-1, 0, 1, 2, 3\}\) and \(B = \{0, 1, 4, 9\}\). How many elements does the subset of \(A \times B\) corresponding to the relation \(R = \{(x, y): |x| < y\}\) have, where \(x \in A\) and \(y \in B\)?
Important Questions from Relations and Functions
If the function of \(f(x)=\dfrac{x}{x-1}\) express f(3x) in terms of f(x)
If \(f(x)-\dfrac{1}{1+2^{1/x}}\) then at x = 0 the function is:
If f(x) is a periodic function and a is a positive real number such that f(x + 2α) + f(x) = 0 for all x ∈ ℝ, then the period of f(x) is:
If \(f(x)=\frac{1}{1+x}\), g(x) = f{f(x)} and h(x) = f[f{f(x)}], then the value of f(x).g(x).h(x) is:
If \(f(x) = {\sin ^{ - 1}}\left[ {\frac{{\sqrt 3 }}{2}x - \frac{1}{2}\sqrt {1 - {x^2}} } \right]\), \(x \in \left[ { - \frac{1}{2},1} \right]\), then f(x) will be
