Problem 111
Question
If the function \(f\) satisfies the relation \(f(x+y)+\) \(f(x-y)=2 f(x) f(y) \forall x, y \in R\) and \(f(0) \neq 0\), then \(f(x)\) is (A) an even function (B) an odd function (C) odd if \(f(x)>0\) (D) neither even nor odd
Step-by-Step Solution
Verified Answer
(A) an even function
1Step 1: Understand the given functional equation
We are given the equation \(f(x+y) + f(x-y) = 2 f(x) f(y)\) for all real numbers \(x\) and \(y\). Additionally, it is mentioned that \(f(0) eq 0\). We need to use these conditions to determine the nature of the function \(f(x)\).
2Step 2: Verify f(0) using y = 0
Substitute \(y = 0\) into the functional equation to find the value of \(f(x)\):\[f(x+0) + f(x-0) = 2 f(x) f(0)\]which simplifies to:\[2f(x) = 2f(x)f(0)\] Dividing both sides by \(2\) gives us:\[f(x) = f(x)f(0)\].Since \(f(x) eq 0 \) generally, divide by \(f(x)\) leading to:\[f(0) = 1\].
3Step 3: Check f(x) at symmetric points
Substitute what we have found: \(x = 0\) in the original equation:\[f(0+y) + f(0-y) = 2f(0)f(y)\] Now substituting \(f(0) = 1\): \[f(y) + f(-y) = 2f(y)\]This simplifies to:\[f(-y) = f(y)\].That means the function \(f(x)\) is even since it satisfies \(f(-x) = f(x)\).
4Step 4: Analyze implications of the results
The equation \(f(-y) = f(y)\) indicates that \(f\) is an even function. This is directly from the definition where an even function satisfies \(f(-x) = f(x)\). Thus, \(f(x)\) is indeed an even function.
Key Concepts
Functional EquationSymmetric PointsReal Numbers
Functional Equation
When we talk about a functional equation, we mean an equation that involves a function and its values at some points. These types of equations are used to explore the properties of the function itself. In the given problem, the functional equation is \( f(x+y) + f(x-y) = 2f(x)f(y) \) for all real numbers \( x \) and \( y \). This equation is not just a simple addition equation; instead, it holds for any values of \( x \) and \( y \), meaning it captures the inherent nature or symmetry of \( f \).
Working through functional equations often involves substituting specific values for variables to derive useful information about the function. For instance, by choosing \( y = 0 \), we glean insights into the function when evaluated at zero. The functional equation thereby becomes a powerful tool in understanding the function's behavior, symmetry, and possible identities.
Working through functional equations often involves substituting specific values for variables to derive useful information about the function. For instance, by choosing \( y = 0 \), we glean insights into the function when evaluated at zero. The functional equation thereby becomes a powerful tool in understanding the function's behavior, symmetry, and possible identities.
Symmetric Points
Symmetric points in mathematics refer to pairs of points that are equidistant from a center point but on opposite sides of it. In the context of the function problem we are solving, symmetric points can be explored by evaluating the function at \( x+y \) and \( x-y \), which is symmetrically equivalent around an axis or point.
By analyzing how the function behaves at these computes, we might recognize a pattern or symmetry in the function. For example, in our solution, we found that \( f(y) + f(-y) = 2 f(y) \). This directly helps identify the symmetry of the function, confirming that it is even. The equation \( f(-y) = f(y) \) highlights that, for any point \( y \), the function value is symmetric with respect to zero, emphasizing its even nature.
By analyzing how the function behaves at these computes, we might recognize a pattern or symmetry in the function. For example, in our solution, we found that \( f(y) + f(-y) = 2 f(y) \). This directly helps identify the symmetry of the function, confirming that it is even. The equation \( f(-y) = f(y) \) highlights that, for any point \( y \), the function value is symmetric with respect to zero, emphasizing its even nature.
Real Numbers
Real numbers are the fundamental elements of analysis in mathematics. They include all the possible numbers along the number line, from negative to positive infinity, and incorporate integers, fractions, and irrational numbers alike. In the exercise, all values of \( x \) and \( y \) are specified to be real numbers, which means the given functional equation holds across the entire spectrum of real values.
This universality stresses the versatility and comprehensiveness of the functional equation, as it applies for any real number choice of \( y \) and \( x \).
This universality stresses the versatility and comprehensiveness of the functional equation, as it applies for any real number choice of \( y \) and \( x \).
- It is vital to understand that by covering all real numbers, the set of solutions or properties we derive from the equation are exhaustive and universally applicable.
- This characteristic of being defined over all real numbers is crucial for identifying properties like evenness or oddness of the function \( f \).
Other exercises in this chapter
Problem 106
If \(\\{x\\}\) and \([x]\) represent fractional and integral part of \(x\), then the value of \([x]+\sum_{r=1}^{2000} \frac{\\{x+r\\}}{2000}\) is (A) \(x\) (B)
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Consider a function \(f(n)\) defined for all \(n \in N\). The function satisfies the following two conditions (i) \(f(1)+f(2)+f(3)+\ldots\) to \(\infty=1\) (ii)
View solution Problem 112
Let \(f: R-\\{2\\} \rightarrow R\) be a function satisfying \(2 f(x)+\) \(3 f\left(\frac{2 x+29}{x-2}\right)=100 x+80 \forall x \in R-\\{2\\}\), then \(f(x)=\)
View solution Problem 113
Let \(g: R \rightarrow R\) be given by \(g(x)=3+4 x\). If \(g^{n}(x)=\) gogo ... \(\operatorname{og}(x)\), then \(g^{\circ}(x)\) (where \(g^{-n}(x)\) denotes in
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