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Find the inverse function in slope-intercept form (mx+b) : f(x)=(4)/(5)x-16 Answer: f^(-1)(x)=

Find the inverse function in slope-intercept form (mx+b) (\mathrm{mx}+\mathrm{b}) :\newlinef(x)=45x−16 f(x)=\frac{4}{5} x-16 \newlineAnswer: f−1(x)= f^{-1}(x)=

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Full solution

Q. Find the inverse function in slope-intercept form (mx+b) (\mathrm{mx}+\mathrm{b}) :\newlinef(x)=45x−16 f(x)=\frac{4}{5} x-16 \newlineAnswer: f−1(x)= f^{-1}(x)=
  1. Understand Inverse Function Concept: Understand the concept of an inverse function. An inverse function essentially reverses the operation of the original function. If f(x)f(x) takes an input xx and produces an output yy, then the inverse function f−1(x)f^{-1}(x) takes yy as an input and produces the original xx as an output.
  2. Write Original Function: Write down the original function.\newlineThe original function is f(x)=45x−16f(x) = \frac{4}{5}x - 16. To find the inverse, we need to switch the roles of xx and yy.
  3. Replace with y: Replace f(x)f(x) with yy. We write the function as y=45x−16y = \frac{4}{5}x - 16. This will make it easier to manipulate the equation to solve for xx.
  4. Swap xx and yy: Swap xx and yy.\newlineTo find the inverse, we switch xx and yy, so we get x=45y−16x = \frac{4}{5}y - 16.
  5. Solve for y: Solve for y.\newlineNow we need to solve the equation x=45y−16x = \frac{4}{5}y - 16 for yy. First, we'll add 1616 to both sides to isolate the term with yy on one side: x+16=45yx + 16 = \frac{4}{5}y.
  6. Multiply by 54\frac{5}{4}: Multiply both sides by 54\frac{5}{4} to solve for yy. To get yy by itself, we multiply both sides by the reciprocal of 45\frac{4}{5}, which is 54\frac{5}{4}: (54)(x+16)=y\left(\frac{5}{4}\right)(x + 16) = y.
  7. Distribute 54\frac{5}{4}: Distribute 54\frac{5}{4} to both terms on the left side.\newlineWe distribute 54\frac{5}{4} to xx and to 1616: (54)x+(54)⋅16=y\left(\frac{5}{4}\right)x + \left(\frac{5}{4}\right)\cdot16 = y.
  8. Simplify the Equation: Simplify the equation.\newlineWe simplify the second term: (54)⋅16=20(\frac{5}{4})\cdot 16 = 20. So the equation becomes (54)x+20=y(\frac{5}{4})x + 20 = y.
  9. Write Inverse Function: Write the inverse function.\newlineNow that we have yy by itself, we can write the inverse function as f−1(x)=54x+20f^{-1}(x) = \frac{5}{4}x + 20.

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