I am not entirely certain of what you mean in all of these problems but I will try to solve them to the best of my ability.
It appears you have asked three seperate questions.
A) What is f(1) when the f(x) = 5x+1
To solve this, you can simply substitute in 1 any time you see an x.
This means
f(x) = 5x+1
becomes
f(1) = 5(1) + 1
From there you need only solve the problem using the order of operations. There is nothing to solve inside parentheses, nor are there exponents, so you can just skip straight to multiplication
f(1) = 5 + .1
And then addition.
f(1)=6
Therefore, if the f(x) = 5x +1, the f(1) = 6
B) If the f(x) =[tex] \frac{2x + 3}{5} [/tex] what is the [tex]f ^{-1} [/tex](3)
I do not have much experience solving problems like this, but I will try to help you.
So the f-1(3) is the inverse function of three. That means that it's basically the same only reversed.The way to find the inverse of f(x) =[tex] \frac{2x + 3}{5} [/tex] would be to substitute f(x) for a y, and then switch the x's and y's, so that it becomes
[tex]x = \frac{2y+3}{5} [/tex]
then simplify and solve that.
5x = 2y+3
5x-3 = 2y
[tex] \frac{(5x-3)}{2} =y [/tex]
[tex] y= \frac{(5x-3)}{2} [/tex]
Then once more substitute the y for f(x), and you have the inverse function of x
f(x) = [tex] \frac{(5x-3)}{2} [/tex]
From there you can just solve for the f(3)
First substitute in the 3,
f(3) =[tex] \frac{(5(3)-3)}{2} [/tex]
Then, follow the order of operations
Parentheses
f(3) = [tex] \frac{(15-3)}{2} [/tex]
f(3) = [tex] \frac{12}{2} [/tex]f(3) = 2
In this case, the f-1(3) = 6 if f(x) =[tex] \frac{2x + 3}{5} [/tex]
However, I am not 100 percent certain that is the correct way to go about solving this problem, so I am also going to try solving it a different way here, in hopes that someone else will know what is correct.
Since ordinarily, when solving for the inverse of a function written as f-1(x), you would switch x and y, when solving the inverse of a function written as f-1(3), perhaps it would be correct to switch 3 and y. I will now attempt to solve this in this way.
If the f(x) =[tex] \frac{2x + 3}{5} [/tex] what is the [tex]f ^{-1} [/tex](3)
So the way I would go about this is to first substitute in the 3, replacing all x's
f-1(3)= [tex] \frac{(2(3) + 3)}{5} [/tex]
Then I would replace f-1(3) with y
y = [tex] \frac{(2(3) + 3)}{5} [/tex]
And switch the y with the tree i just substituted in as an x
3= [tex] \frac{(2(y) + 3)}{5} [/tex]
Then I would only need to solve the problem, first isolating the variable, and then simplifying as far as possible.
In the process of isolating the y, I would first multiply by five to remove the fraction,
15= 2y+3
Then subtract 3
12=2y
and divide by 2 to remove the coefficient
6=y
and reverse the two sides for the sake of convenience.
y=6
Then I would change y back to f-1(3) to keep things consistent.
f-1(3) = 6.
Either way, you get the same answer, I am just uncertain as to which is the technically correct method.
C)3y − 7 = y + 5
This is a fairly simple problem, and the solution can be found by moving the variables and numbers so that each is on its own side of the equal sign, then multiplying or dividing until it is further simplified.
First, rewrite the problem
3y − 7 = y + 5
Then, move the numbers so that they are all on one side. In this case, I will do so by adding 7
3y =y +5 +7
3y =y+12
Then move the variables so they are all on the same side. In this case, I am subtracting y from both sides.
3y -y = 12
2y = 12
And finally, I will divide by 2 to further simplify
y = 12
3y − 7 = y + 5 simplified, is y = 12
