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Obtaining a function from an equation

Equations and functions are not the same thing, but they can be related in several ways. In this video, we obtain a function from an equation. The function represents the same relationship between the quantities in the equation.

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  • primosaur ultimate style avatar for user GodIsREAL
    Hi, there is this one problem in the exercise "Write function rules from equations" that I am having a hard time with. It wants me to enter the greek letter upsilon as part of the answer, but I don't know how. How can I do that??
    (82 votes)
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    • male robot hal style avatar for user Feraru Silviu Marian
      The input is b, so, f(b), the output is a, so f(b)=a, whatever input b we plug into our function, it's gonna output a.
      Therefore, to satisfy the equation we need to solve the equation in terms of a, and then just replace the a in f(b)=a, and that's our function, bellow is a summary of the steps.
      f(b)=a // whatever b we input, the function outputs a
      4a+7b = -52 // this is the equation our a has to satisfy
      a = -13-(7/4)*b // therefore we solve for a, so the equation stands true
      f(b) = -13-(7/4)*b // we replace the a in f(b)=a with whatever a equals to
      (13 votes)
  • orange juice squid orange style avatar for user God Serena
    I am doing the exercises now but I haven't gotten a single one right. I don't think Sal gave an adequate enough explanation. He only did a couple examples and when I look at the hints they seem lazy and not very in depth... XD what do I do?
    (36 votes)
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    • mr pink green style avatar for user David Severin
      So here is one of the practice problems:
      7m+2=6n−5 what is f(m)?
      Since the independent variable is m indicated by being in the functional parenetheses, we must solve for the dependent variable, and the only one left is n, so lets solve for n
      6n - 5 = 7m + 2 just flipped sides of equation
      6n = 7m + 2 + 5 moved the -5 by adding 5 on both sides
      6n = 7m + 7 combined like terms
      n = 7/6 m + 7/6 divided all terms by 6
      f(m) = 7/6m +7/6 substituted functional notation with equation notation
      Does this help?
      (45 votes)
  • blobby green style avatar for user Sephora Ndiaye
    Why would you solve for A if you are trying to figure out a formula for B ? In other words trying to get B by it self. For this problem I subtracted 4a by itself and -52 which then simplified to 7b= -52-4a / 7 which simplified even more to which b = -52-4a / 7
    (18 votes)
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  • aqualine ultimate style avatar for user Keshav Agrawal
    So basically -13 - (7/4b) is the same as a, which is the same as f(b). Am I right?
    (11 votes)
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    • hopper jumping style avatar for user Michael Laman
      Yes, you are correct.

      When f(b) is given a "b" value we get an "a" value which is the same as "f(b)".

      For example if we're given the "b" value of 0, then f(0) = -13 and that is what "a" would be if "b" is zero. "f(b)" is a more concise way to say this since it says what value of "b" was used to get the "f(b)" value - to get a particular "a" value.

      I hope this helps.
      (16 votes)
  • male robot donald style avatar for user Kush K.
    In school, I have a teacher that doesn't explain anything. I am a 6th grader so in Algebra 1 and I am having a lot of trouble with functions. I can solve functions but creating a linear function equation is difficult for me. Could anyone help explain it to me in an easy way that I could understand?
    (16 votes)
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  • piceratops seed style avatar for user Robert Fascaldo
    I was starting to understand this until I ran into a problem on the exercises.
    a - 7 = 3(b+2)

    I was able to get a by itself

    a = 3(b+2)+7

    in the hints it said it would simplify down to

    a = 3b + 13.

    My question is how do you get 13?
    (1 vote)
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  • starky seedling style avatar for user Gayathri sudarsan
    so technically an equation in two variables is a function and equation in one variable is just an equation
    (4 votes)
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    • stelly blue style avatar for user Kim Seidel
      Not necessarily.
      A function is a relationship where each input value (X) will create only one output value (Y). Basically, a single input value, can't create 2 different output values.

      Any equation with one or two variables that meet this definition would be a function.
      y = 5 is the equation for a horizontal line. It is a function where all values of X have a y-value = 5. Yet it has one variable.
      x = 5 is the equation for a vertical line. It is not a function because in this situation, the input value (x=5) has an infinite number of output values.
      All other equations of lines (Ax + By = C) are functions because the meet the definition of a function.
      But, the equation of a circle, which also uses 2 variables, would not be a function because each input value (x) creates 2 output values (y).

      If an equation is not a function, sometimes we can still change it into a function if we restrict its inputs or outputs.

      Hope this makes sense.
      (9 votes)
  • blobby green style avatar for user jollyjelly
    this topic definitely needs more explanation !!
    (6 votes)
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  • blobby green style avatar for user 24kpunkdogs
    What? why are we solving for a? why not b? where is the input for the function here? what does solving for a have to do with this f(b). How is f outputting a? im so confused
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      If the problem is asking you to create function f(b), then it is telling you that the input value is defined by "b", not "a". So, "a" becomes the variable that you isolate and switch to f(b).

      If you solved for "b", then the input value would need to be "a" and the problem didn't ask you to create f(a).

      Hope this helps.
      (9 votes)
  • starky tree style avatar for user WitcherVr
    this is really hard
    (5 votes)
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    • stelly blue style avatar for user Kim Seidel
      By now, you should have experience solving equations for "y" to change a linear equation into slope-intercept form. Use those skills.

      The 1st step is to solve the equation for "y" (isolate "y" by itself on one side of the equation.

      The 2nd step is to replace "y" with f(x), or whatever name you were asked to use for your function.

      Hope this helps.
      (2 votes)

Video transcript

For a given input value b, the function f outputs a value a to satisfy the following equation 4a plus 7b is equal to negative 52. So for a given input b, the function f, the function f will output an a that satisfies this relationship right over here for the a and the b. Write a formula for f of b in terms of b. So we want to do, we just want to solve for -- if we're given a "b", what "a" does that imply that we have to output? Or another way to think about it is -- let's just solve for a, or we could think about a as being a function of b. So let's write this. So we have 4a plus 7b -- is equal to negative 52. So I can solve for a in terms of b, that any b that I have -- Let's say these b's are on the right hand side I can put it in. I can substitute that value for b and I can just solve for a. I can solve for a that needs to be outputted. So let's do that. Let's solve for a. So I want to get all the a on I wanna just have an a leftover on the left hand side, and have everything else on the right hand side including the b's. So let's get rid of this b on the left hand side. And I can do that by subtracting 7b. Of course I wanna do that to both sides. I can't just do an equation and do an operation only on one side like that. So let's subtract, and we are left with we are left with -- the 7b's add up to zero. 7b minus 7b. We're left with 4a is equal to negative 52 minus 7b, minus 7b. Now, to isolate the a here, just to have an a here instead of 4a, we can divide both sides by 4. We can divide both sides by 4. So I'm gonna divide everything by 4. And on the left hand side, we got our goal. We are left with an a is equal to -- Now what's negative 54 divided by -- What is the negative 52 divided by 4? So let's think about it. 52 is 40 plus 12. 40 divided by 4 is 10. 12 divided by 4 is 3. So it's gonna be 13. Negative 13. So it's negative 13 minus 7/4 b minus 7/4 b. So given a "b", if you give me a "b", I can put that value right over here, and I can calculate what the corresponding a needs to be in order to satisfy this relationship. So if I want a formula for f of b in terms of b, I can say, look, you give me a "b", the output of our function, which is f of a. The output of our function is going to be -- is going to be negative 13 minus 7/4 b, because the output of our function needs to be an "a" that will satisfy -- that will satisfy this equation up here. So hopefully that helped.