Chapter 9.9, Problem 45PPS

To determine

To describe sometime, always or never true.

Answer to Problem 45PPS

Sometimes

Explanation of Solution

Given information:

  $f(x)$ and $g(x)$are linear functions

  $(f+g)(x)$is all real numbers.

Suppose we have two functions, $f(x)$ and $g(x)$ . We can define the difference of these two functions by,

  $(f-g)(x)=f(x)-g(x),$

where x is in the domain of both f and g .

For example, we can substract the functions $f(x)=x^2-1$and $g(x)=2x^2+3$ as,

  $(f-g)(x)=f(x)-g(x)$

  $=x^2-1-(2x^2+3)$

On solving,

  $=-x^2-4.$

The domain of $(f-g)(x)$ consists of all x -values that are in the domain of both f and g . In this example, f and g both are quadratic function and their substraction is also a quadratic function, therefore, $(f-g)(x)$has linear function sometimes.