Need help with honors thing for algebra 1. Part 1. Using the two functions listed below, insert numbers in place of the letters a, b, c, and d so that f(x) and g(x) are inverses. http://prntscr.com/buegvy Part 2. Show your work to prove that the inverse of f(x) is g(x). Part 3. Show your work to evaluate g(f(x)).
Part 4. Graph your two functions on a coordinate plane. Include a table of values for each function. Include 5 values for each function. Graph the line y = x on the same graph. Can someone help me and show me how to do this?
for part A write \[ y = \frac{x+a}{b}\] to find the inverse, rename y to x and x to y (i.e. "switch x and y") can you do that step ?
So part one would = to x = y+a/b?
@phi
yes but you should use parens x= (y+a)/b to show b is dividing both y and a next step: multiply both sides by b (that means write "b" on both sides) then simplify if you know how.
Like that? http://prntscr.com/buespl
yes. the b/b on the right side simplifies to 1 x b = y+a we can drop the parens on (y+a) now add -a to both sides and simplify (the right side)
you should get xb-a = y or y= b x - a ( x*b it's the same as b*x)
Is that part 3?
if you compare that to the formula they gave you g(x) = c x - d y = b x - a it looks like we want c to be "b" and d to be "a" g(x) = b x -a f(x) = (x+a)/b those are inverses. they want you to pick numbers for a and b I would pick nice numbers.
I am kind of confused :/
ok, that means you are missing some idea. did you see how we found y = b x - a using algebra ?
Yes. Is that the inverse?
yes, we followed a "procedure" to find the inverse: 1) switch x and y 2) solve for y they said that f(x) = (x+a)/b g(x) = c x - d where g(x) is the inverse. we found the inverse was y = b x - a
Oh ok. Now replace b and a with, lets say, 1 and 2. So y = 1x - 2, would we do it for the inverse or the inverse and f(x)?
for their g(x) = c x - d to match what we found, it must be c is really b and d is "a" so the two pair are f(x) = (x+a)/b g(x) = b x - a
Now replace b and a with, lets say, 1 and 2. ok, and using those numbers, you can write down *both* f(x) and g(x)
what do you get for f(x) and g(x) using a=2 and b=1 ?
f(x) = x + 2 (just removed the /1 :P) g(x) = x - 2
now we do part B show f(x) and g(x) are inverses.
http://prntscr.com/buf0y0 should my answer look like that for now?
I just wrote what we did, is that enough?
no. part 2 should be first, and then write: let a=2 and b=2 then write what you have in part 1. all of that is to find f(x) and its inverse. i.e. part A
so part 1 is part 2 and part 2 is part 1?
now for Part 2. Show your work to prove that the inverse of f(x) is g(x). start with the definitions: f(x) = x + 2 g(x) = x - 2 next, write f( g(x) ) = in the definition of f, everywhere you see x, erase the x, and write in g(x)
**so part 1 is part 2 and part 2 is part 1? *** all of that is Part 1. But you wrote it up in the wrong order.
Alright, that good? http://prntscr.com/buf4jc
yes, but right before f(x)= x+2 you should say g(x) = b x - a and then say: let a=2 and b =1 so that people know how you got f(x)=x+2 and g(x) = x-2
Sorry for being such a pane. http://prntscr.com/buf60g
Pain, omg my brain is not in the right place today.
that looks pretty good. But the last line (for Part 2) were instructions for you to do that to figure out what f( g(x) ) = is equal to. in other words, write f( g(x) ) = write the definition of f(x) here, but replace x with g(x)
the definition of f(x) is x+2
So, when you say definition do you mean words or what it =? Because that would be f(x-2)
OH OK, I GET IT NOW. 1 second.
f( g(x) ) = g(x) + 2 ?
yes, next, on the right side, replace g(x) with its definition
Step 2 still? x - 2 + 2, or just x. So the answer would be f(g(x)) = x?
yes, but it's good to show the steps: f( g(x) ) = g(x) + 2 f( g(x) ) = x - 2 + 2 f( g(x) ) = x now we do the same for g( f(x) ) = ??? care to try ?
Alright, Part 2 or 3? The answer would be: g(f(x)) = f(x) - 2 g(f(x)) = x + 2 -2 g(f(x)) = x?
yes, but (obviously) no ? at the end this is still Part 2 you now say f(g(x)) = x = g(f(x)) which means f(x) and g(x) are inverses.
for Part 3, you already did that work in Part 2
yes
How do I do part 4 :/
Part 4. Graph your two functions on a coordinate plane. Include a table of values for each function. Include 5 values for each function. Graph the line y = x on the same graph.
do you have to do it by hand ? you need graph paper
I dont have graph paper, and this is an online course... WAT
you could use https://www.desmos.com/calculator
How would I graph the points?
They want a table with 5 points I would use x= -2 , -1 , 0, 1, 2 and figure out f(x) and g(x) for each of those x values
Hm, I think I got this one second.
http://prntscr.com/bufk46 what would I do next?
I would make the first column f(x) and the second g(x) to find the values for f(x) you figure out f(-2), f(-1), etc to, for example, find f(-2), you write the definition of f(x), but replace with x with -2 then simplify.
btw, you could make your table in any order, but it makes more sense to order the x's from smallest (-2) to biggest (+2)
So would that make my first point -2, -2?
yes, but they want you to "evaluate" f(-2) to get a number for example f(x) = x+2 (that is the definition) f(-2) = -2 + 2 (f(-2) means replace the x with -2) f(-2) = 0 (simplify)
http://prntscr.com/bufwoh What would I do next @phi
@welshfella Was I supposed to make C b and D a?
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