Question

Combine all the following functions and evaluate when necessary.

Answer 1

Answer:

$g(x) - f(x)=-x-10$

$f(x) +h(x)=8x+8$

$g(x) * h(x)=10x^2-24x$

$(h-g)(4)=0$

Step-by-step exp

lanation:

1) The subtraction of functions  $g(x) - f(x)$ can be written by replacing them with expressions on the variable "x" they give us for their definition:

$g(x) - f(x)=(5x-2)-(6x+8)=5x-2-6x-8=-x-10$

Notice that when we did the replacement with the functions' expression, we use parentheses. This is a very important and safe step, that will help us be careful about subtractions especially when there are several algebraic terms involved. As we see a negative sign before a parenthesis, we understand that at removing this grouping symbol, all the signs of the terms inside must be flipped. A positive sign in front of the parentheses will not change the signs of the terms inside it.

When then proceed to combine like terms to get the final expression:

$g(x) - f(x)=-x-10$

2) With this addition , we follow the same method as before, and the process is more straight forward because there are no subtractions and therefore no negative signs before grouping symbols that could change the signs of the terms involved:

$f(x) +h(x)=(6x+8)+(2x)= 6x+8+2x=8x+8$

3) We proceed in a similar way for the product of these two functions, making sure that we use distributive property to remove parentheses:

$g(x) * h(x)= (5x-12)(2x)=10x^2-24x$

4) to evaluate the difference of two functions at a given value, we can use a shortcut of evaluating each function at the point in question, and then subtracting the result, instead of finding first the function form for the subtraction:

$h(4)=2*(4)=8\\g(4)=5*(4)-12=20-12=8\\h(4)-g(4)=8-8=0$

If we find the subtraction of functions first:

$(h-g)(x)=h(x)-g(x)=(2x)-(5x-12)=2x-5x+12=-3x+12$

and then we evaluate what this new function gives at the value x=4 we should get the same answer as before:

$(h-g)(4)= -3(4)+12=-12+12=0$