+0

+10
226
5
+701

1) Define g by g(x) = 5x - 4. If g(x) = f^-1(x) - 3 and f^-1(x) is the inverse of the function f(x) = ax+b, find 5a + 5b.

So far, I've found the inverse of f, which is (x-b)/a. The equation would be $5x - 4 = (x-b)/a - 3$. Please help me find $a$ and $b$ to get $5a + 5b$. Thank you so much!

(I completely do not get the second question...I don't understand what the brackets mean)

2) Let g(x) be a function piecewise defined as $f(x) = \left\{ \begin{array}{c l} -x & x\le 0 \\ 2x-41 & x>0 \end{array} \right..$

If $a$ is negative, find $a$ so that $g(g(g(10.5))) = g(g(g(a)))$. Express your answer as a decimal.

(If you can only answer one of the two questions, that is fine!)

Thank you so much!

Nov 18, 2018

#1
+101181
+3

f(x)   =  ax + b    for f(x), write y

y = ax + b      subtract b rom both sides

[y - b]  =  ax   divide both sides by a

[ y - b] / a  =  x      "swap' x and y

[x - b] / a  = y

(1/a)x - b/a  = y  =   f-1(x)

So g(x)  = f-1(x) - 3  =   (1/a)x - b/a - 3  =  5x - 4

This implies that

(1/a) = 5       and    -b/a - 3  = 4

So

a = (1/5)      and

-b / (1/5) - 3  =  -4     add 3 to both sides

-b / (1/5)  =  -1

-b =  -1 (1/5)

-b = -1/5

b = 1/5

So

5a + 5b  =

5(1/5) + 5(1/5)  =    1 + 1   =   2

Nov 18, 2018
#2
+701
0

Thanks so much, CPhill! I fully understand the solution now!

Solution to second question is $$a = -30.5$$.

PartialMathematician  Nov 18, 2018
edited by PartialMathematician  Dec 9, 2018
#3
+101755
+2

Hi PartialMathematician, welcome to the forum.

In a couple of years time are you going to change your name to FullMathematician ?  LOL

On a serious note:

The stuff between the dollar signs is LaTex code.

If you delete the dollar signs you can past the rest into the LaTex box that will appear when you click [LaTex] in the ribbon.

After you hit ok it will present properly.

Note:

Just delete the stuff that is in the box when you open it.

Nov 18, 2018
#4
+101181
+1

LOL!!....when you think about it....most of us are all just "Partial Mathematicians"  ...!!!!

CPhill  Nov 18, 2018
#5
+101755
+2

Yes, that is right :)

I guess he/she can keep the name :))

Melody  Nov 18, 2018