Math  /  Algebra

QuestionFind the inverse function in slope-intercept form (mx+b)(\mathrm{mx}+\mathrm{b}) : f(x)=54x5f(x)=-\frac{5}{4} x-5

Studdy Solution

STEP 1

What is this asking? We're taking a linear function and flipping it around to find its inverse, which will also be a line, and we want that inverse in slope-intercept form. Watch out! Don't forget to swap xx and yy *before* you start rearranging the equation!
Also, keep track of those negative signs – they can be sneaky!

STEP 2

1. Rewrite in terms of x and y
2. Swap x and y
3. Solve for y
4. Rewrite in slope-intercept form

STEP 3

We **start** with our function: f(x)=54x5f(x) = -\frac{5}{4}x - 5 Let's rewrite this equation using yy instead of f(x)f(x), because it's easier to work with when finding the inverse.
So, we have: y=54x5y = -\frac{5}{4}x - 5

STEP 4

To find the inverse, we **swap** xx and yy.
This reflects the function across the line y=xy = x.
It's like looking at the function's mirror image!
So, our equation becomes: x=54y5x = -\frac{5}{4}y - 5

STEP 5

Now, we want to **solve** for yy to get the inverse function.
First, let's add 5 to both sides of the equation to isolate the term with yy: x+5=54yx + 5 = -\frac{5}{4}y

STEP 6

Next, we **multiply** both sides by 45-\frac{4}{5} to get yy by itself.
Remember, multiplying by 45-\frac{4}{5} is the same as dividing by 54-\frac{5}{4}.
We're doing this to make the coefficient of yy equal to **one**: 45(x+5)=45(54y)-\frac{4}{5} \cdot (x + 5) = -\frac{4}{5} \cdot \left( -\frac{5}{4}y \right)

STEP 7

Now, let's **simplify**: 45x455=y-\frac{4}{5}x - \frac{4}{5} \cdot 5 = y 45x4=y-\frac{4}{5}x - 4 = y

STEP 8

Our equation is already in slope-intercept form (y=mx+by = mx + b), where mm is the slope and bb is the y-intercept!
So, the **inverse function** is: y=45x4y = -\frac{4}{5}x - 4 We can also write this using inverse function notation: f1(x)=45x4f^{-1}(x) = -\frac{4}{5}x - 4

STEP 9

The inverse function in slope-intercept form is f1(x)=45x4f^{-1}(x) = -\frac{4}{5}x - 4.

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