[will mark as brainliest] find the inverse of the given function. f(x) = -1/2√x + 3, x ≥ -3 f^-1 (x) = x^2 - for x ≤

The answer is:

The inverse of a function is another function, , with the following property:



In other words, the inverse of a function does exactly "the opposite" of what the original function does, and so if you compute them both in sequence you return to the starting point.

Think for example to a function that doubles the input, , and one that halves it: . Their composition is clearly the identity function , since you consider "twice the half of something", or "half the double of something".

In general, to invert a function , you have to solve the expression for , writing an expression like . If you manage to do so, then is the inverse of .

In your case, you have



Multiply both sides by to get



Square both sides to get



Finally, subtract 3 from both sides to get



Since the name of the variables doesn't really have a meaning, you can say that the inverse function is



As for the domain of the inverse function, remember what we said ad the beginning: if the original function goes from set A (domain) to set B (codomain), then the inverse function goes from set B (domain) to set A (codomain). This means that the inverse function is defined on an element in B if and only if that element belongs to the range of the original function, i.e. the set of the elements of the codomain such that there exists . So, we need the range of .

We know that the range of is . When you transform it to you simply translate the graph horizontally, so the range doesn't change. But when you multiply the function times you affect both extrema of the range, turning it into , which you can simply write as

answer: haha (: i’ll go see what it is

step-by-step explanation:

starting with:    

5 + x - 12 = 2x - 7   

5 + x - 12 - x = 2x - 7 - x (subtract x from both sides)   

5 - 12 = x - 7 (combine and cancel x's from both sides)   

5 - 12 + 7 = x - 7 + 7 (add 7 to both sides)   

0 = x (add numbers together on both sides)       

so in order for the equation to be true, x must be 0.   

trying x = -0.5   

5 + x - 12 = 2x - 7 (original equation)   

5 + -0.5 - 12 = 2(-0.5) - 7 (plug in value of x)   

5 + -0.5 - 12 = -1 - 7 (did multiplication)   

-7.5 = -8 (did addition on both sides)       

since -7.5 is obviously not equal to -8, the value of -0.5 for x is wrong.   

trying x = 0   

5 + x - 12 = 2x - 7 (original equation)   

5 + 0 - 12 = 2(0) - 7 (plug in value of x)   

5 + 0 - 12 = 0 - 7 (did multiplication)   

-7 = -7 (did addition on both sides)       

since -7 equals -7, that demonstrates that the value of 0 for x is correct.   

trying x = 1   

5 + x - 12 = 2x - 7 (original equation)   

5 + 1 - 12 = 2(1) - 7 (plug in value of x)   

5 + 1 - 12 = 2 - 7 (did multiplication)   

-6 = -5 (did addition on both sides)       

since -6 is obviously not equal to -5, the value of 1 for x is wrong.