Question

Some functions \(f\) have the property that $$ f(a+b)=f(a)+f(b) $$ for all real numbers \(a\) and \(b .\) Which of the following functions have this property? (a) \(h(x)=2 x\) (b) \(g(x)=x^{2}\) (c) \(F(x)=5 x-2\) (d) \(G(x)=\frac{1}{x}\)

Short answer

Only \( h(x) = 2x \) satisfies the property.

Step-by-step solution

- Define the property

The problem asks to verify which functions satisfy the property \[ f(a+b) = f(a) + f(b) \] for all real numbers \(a\) and \(b\). This property is known as additivity.

- Test function h(x)

Let \( h(x) = 2x \). Substitute \(a\) and \(b\) into the function: \[ h(a+b) = 2(a+b) \] Calculate \( h(a) + h(b) \): \[ h(a) + h(b) = 2a + 2b \] Compare both expressions: \[ 2(a+b) = 2a + 2b \] Both sides are equal. Therefore, \( h(x) = 2x \) satisfies the property.

- Test function g(x)

Let \( g(x) = x^2 \). Substitute \(a\) and \(b\) into the function: \[ g(a+b) = (a+b)^2 \] Expand the expression: \[ (a+b)^2 = a^2 + 2ab + b^2 \] Calculate \( g(a) + g(b) \): \[ g(a) + g(b) = a^2 + b^2 \] Compare both expressions: \[ a^2 + 2ab + b^2 eq a^2 + b^2 \] Both sides are not equal. Therefore, \( g(x) = x^2 \) does not satisfy the property.

- Test function F(x)

Let \( F(x)=5x-2 \). Substitute \(a\) and \(b\) into the function: \[ F(a+b) = 5(a+b) - 2 \] Calculate \( F(a) + F(b) \): \[ F(a) + F(b) = (5a - 2) + (5b - 2) = 5a + 5b - 4 \] Compare both expressions: \[ 5(a+b) - 2 eq 5a + 5b - 4 \] Both sides are not equal. Therefore, \( F(x) = 5x - 2 \) does not satisfy the property.

- Test function G(x)

Let \( G(x) = \frac{1}{x} \). Substitute \(a\) and \(b\) into the function: \[ G(a+b) = \frac{1}{a+b} \] Calculate \( G(a) + G(b) \): \[ G(a) + G(b) = \frac{1}{a} + \frac{1}{b} = \frac{a+b}{ab} \] Compare both expressions: \[ \frac{1}{a+b} eq \frac{a+b}{ab} \] Both sides are not equal. Therefore, \( G(x) = \frac{1}{x} \) does not satisfy the property.

Key concepts

Linear Functions

Linear functions are the simplest type of functions and play a key role in understanding more complex concepts. For a function to be linear, it must have the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants. A linear function produces a straight line when graphed. One special case of a linear function is when the intercept \( b \) is zero, giving us a form \( f(x) = mx \). This is particularly interesting because these functions maintain the additivity property.

Functional Equations

Functional equations are mathematical expressions involving functions and their values at particular points. The equation discussed in the exercise, \[ f(a+b) = f(a) + f(b) \], is a well-known example. This equation is specific to functions that are additive. Many problems in calculus and analysis require verification of whether a given function satisfies a specific functional equation. By testing different functions, we can determine whether they maintain this additive property, helping clarify deeper relationships between variables.

Property of Functions

Properties of functions describe how functions behave and interact under various operations. The additivity property investigated here means that the function's value at the sum of two inputs equals the sum of the function's values at those inputs, i.e., \( f(a+b) = f(a) + f(b) \). Each function must be tested to observe whether it maintains this behavior:

• For \( h(x) = 2x \), \( h(a+b) = 2(a+b) \), and indeed, \( 2a + 2b \).
• For \( g(x) = x^2 \), expanding \( (a+b)^2 \) yields extra terms, hence does not satisfy additivity.
• For \( F(x) = 5x - 2 \), the extra constant breaks the property.
• For \( G(x) = \frac{1}{x} \), the function fails to satisfy the additivity property when summed.

Algebraic Properties

Understanding algebraic properties is essential for analyzing functions and their behaviors. Algebraic properties include operations such as addition, multiplication, and how these operations affect function behavior. In the context of functional equations, particularly the additivity property, careful algebraic manipulation is required to verify if the functions hold true to \( f(a+b) = f(a) + f(b) \). The careful step-by-step algebraic calculations allow us to break down complex expressions and make comparisons, as shown in the exercise. By simplifying and expanding, we can see whether both sides of the equations match and thus verify these properties.