Question

suppose that the functions f and g are defined for all real numbers x as follows.
$f(x)=x-3$
$g(x)=2x-6$
write the expressions for $(g - f)(x)$ and $(g cdot f)(x)$ and evaluate $(g + f)(4)$.
$(g - f)(x) = $
$(g cdot f)(x) = $
$(g + f)(4) = $

Explanation:

Step1: Find \((g - f)(x)\)

To find \((g - f)(x)\), we use the formula for the difference of two functions: \((g - f)(x)=g(x)-f(x)\). Substitute \(f(x)=x - 3\) and \(g(x)=2x - 6\) into the formula:

$$
LATEXBLOCK0
$$

Step2: Find \((g\cdot f)(x)\)

To find \((g\cdot f)(x)\), we use the formula for the product of two functions: \((g\cdot f)(x)=g(x)\cdot f(x)\). Substitute \(f(x)=x - 3\) and \(g(x)=2x - 6\) into the formula:

$$
LATEXBLOCK1
$$

Step3: Find \((g + f)(4)\)

First, find the formula for \((g + f)(x)\) using the sum of two functions: \((g + f)(x)=g(x)+f(x)\). Substitute \(f(x)=x - 3\) and \(g(x)=2x - 6\) into the formula:

$$
LATEXBLOCK2
$$

Now, evaluate \((g + f)(x)\) at \(x = 4\):

$$
LATEXBLOCK3
$$

Answer:

\((g - f)(x)=\boldsymbol{x - 3}\)
\((g\cdot f)(x)=\boldsymbol{2x^{2}-12x + 18}\)
\((g + f)(4)=\boldsymbol{3}\)