Question

suppose that the functions $g$ and $h$ are defined for all real numbers $x$ as follows.
$g(x)=4x-4$
$h(x)=3x$
write the expressions for $(h - g)(x)$ and $(h\cdot g)(x)$ and evaluate $(h + g)(1)$.
$(h - g)(x) = \square$
$(h\cdot g)(x) = \square$
$(h + g)(1) = \square$

Explanation:

Step1: Define $(h-g)(x)$

$(h-g)(x) = h(x) - g(x)$

Step2: Substitute functions

$(h-g)(x) = 3x - (4x - 4)$

Step3: Simplify the expression

$(h-g)(x) = 3x - 4x + 4 = -x + 4$

Step4: Define $(h \cdot g)(x)$

$(h \cdot g)(x) = h(x) \cdot g(x)$

Step5: Substitute functions

$(h \cdot g)(x) = 3x(4x - 4)$

Step6: Expand and simplify

$(h \cdot g)(x) = 12x^2 - 12x$

Step7: Define $(h+g)(1)$

$(h+g)(1) = h(1) + g(1)$

Step8: Evaluate $h(1)$ and $g(1)$

$h(1)=3(1)=3$, $g(1)=4(1)-4=0$

Step9: Sum the values

$(h+g)(1) = 3 + 0 = 3$

Answer:

$(h - g)(x) = -x + 4$
$(h \cdot g)(x) = 12x^2 - 12x$
$(h + g)(1) = 3$