Question

Let f(x) = x + 1 and G(x)=1/x What is the range of (F*G)(X)

Answer 1

ANSWER

$y \ne1$

EXPLANATION

The given functions are

$f(x) = x + 1$

and

$g(x) = \frac{1}{x}$

We want to find

$(f \times g)(x)$

We use function properties to obtain:

$(f \times g)(x) = f(x) \times g(x)$

$(f \times g)(x) = (x + 1) \times \frac{1}{x} = \frac{x + 1}{x}$

There is a horizontal asymptote at:

$y = 1$

Let

$y = \frac{x + 1}{x}$

$xy = x + 1$

$xy - x = 1$

$x(y - 1) = 1$

$x = \frac{1}{y - 1}$

The range is

$y \ne1$

Or

$( - \infty ,1) \cup(1, \infty )$

Answer 2

$f(x)=x+1 \\g(x)=\dfrac{1}{x} \\(f\cdot g)(x)=(x+1)\dfrac{1}{x} \\(f\cdot g)(x)=\underline{\dfrac{x+1}{x}} \\ \\0=\dfrac{x+1}{x} \\0=\dfrac{x}{x}+\dfrac{1}{x} \\0=1+\dfrac{1}{x} \\-1=\dfrac{1}{x} \\-x=1 \\x=1$