Question

1. ★ writing tell whether the sum of two power functions is sometimes, always, or never a power function. explain your reasoning.

Explanation:

A power function has the form \( f(x)=ax^{n} \), where \( a \) is a constant and \( n \) is a real number. Let's consider two cases:

1. Case 1: Same exponent \( n \)

If we have \( f(x)=ax^{n} \) and \( g(x)=bx^{n} \), their sum is \( (a + b)x^{n} \), which is a power function (since \( a + b \) is a constant and \( n \) is real). For example, \( f(x)=2x^{3} \) and \( g(x)=3x^{3} \) sum to \( 5x^{3} \), a power function.

1. Case 2: Different exponents

If \( f(x)=2x^{2} \) and \( g(x)=3x^{3} \), their sum is \( 2x^{2}+3x^{3} \), which cannot be written as \( cx^{k} \) for a single constant \( c \) and real number \( k \) (since the exponents differ). Thus, it is not a power function.

Since the sum is a power function in some cases (same exponents) but not in others (different exponents), the sum of two power functions is sometimes a power function.

Answer:

sometimes