QuestionSolve the equation $f(x p + u) = j + a$ for $f$ and $p$ . Find: $f =$ and $p =$ .

Studdy Solution

STEP 1

Assumptions1. The given equation is $f(xp+u)=j+a$ . We are asked to solve for $f$ and $p$
3. All symbols represent real numbers4. The function $f$ is invertible

STEP 2

First, we will solve for $f$ . To do this, we need to isolate $f$ on one side of the equation. Since $f$ is a function applied to the term $(xp+u)$ , we can use the inverse function of $f$ to both sides of the equation to isolate $f$ .
$f^{-1}(f(xp+u))=f^{-1}(j+a)$

STEP 3

The inverse function $f^{-1}$ cancels out $f$ on the left side of the equation, leaving us with $xp+u=f^{-1}(j+a)$

STEP 4

Now, we have isolated the function $f$ in terms of the other variables. So, the solution for $f$ is $f=f^{-1}(j+a)-u/xp$

STEP 5

Next, we will solve for $p$ . We start from the equation obtained in3 $xp+u=f^{-1}(j+a)$

STEP 6

To isolate $p$ , we first subtract $u$ from both sides of the equation $xp=f^{-1}(j+a)-u$

STEP 7

Then, we divide both sides of the equation by $x$ $p=\frac{f^{-1}(j+a)-u}{x}$ So, the solution for $p$ is $p=\frac{f^{-1}(j+a)-u}{x}$

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