How Do Absolute Values Affect Solving Function Equations?

[PhysicslyDSBL]

Homework Statement

I do not see how the two equations in each example are related, what should I do with them? (the l's are absolute value brackets):

a) Let g(x) = 3x - 3 + l x+5 l. Find all values of a which satisfy the equation:

g(a) = 2a +8


b) Let h(x) = l x l - 3x + 4. Find all solutions to the equation :

h(x - 1) = x - 2


I know how to find an equation of the following:

x + l 2x-1 l, find f(x) = 8

x + l 2x-1 l = 8

l 2x-1 l = 8-x

2x-1 = 8-x or -2x+1= 8-x
x=3 x=-7

 

[Mark44]

For part a, you have the formula for g(x), so g(a) = 3a -3 + |a + 3|

Set that expression equal to 2a + 8. Isolate the absolute value expression on one side, and keep in mind that |x| = x if x >= 0 and |x| = -x if x < 0.

 

[Architeuthis Dux]

I would suggest that you first review the concepts of absolute values and functions. Absolute values represent the distance of a number from zero on a number line, and functions are mathematical relationships between two variables. In the given examples, the absolute values are being used in combination with functions to solve equations.

For example, in the first equation, g(x) is a function of x which includes an absolute value of x+5. This means that the value of x+5 can be positive or negative, and the absolute value ensures that it is always positive. To solve the equation g(a) = 2a + 8, you will need to plug in different values of a and see which ones satisfy the equation. This is because the absolute value makes it difficult to solve for a single value of a.

In the second equation, h(x) is a function of x which includes an absolute value of x. This means that the value of x can be positive or negative, and the absolute value ensures that it is always positive. To solve the equation h(x-1) = x-2, you will need to plug in different values of x and see which ones satisfy the equation. Again, the absolute value makes it difficult to solve for a single value of x.

In the third equation, the absolute value is being used to find the value of x that will make the function equal to 8. By setting the function equal to 8, you are essentially finding the x-values where the function crosses the horizontal line at y=8. This can be solved by setting the two possible equations (2x-1=8-x and -2x+1=8-x) equal to each other and solving for x.

Overall, understanding the concept of absolute values and how they are used in combination with functions is crucial in solving equations that involve them. I would suggest reviewing the concept and practicing more examples to gain a better understanding.