Math / Algebra

QuestionIf $f(x)=-5 x+1$ , then $f^{-1}(x)=$ $\square$
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Studdy Solution

STEP 1

1. We are given the function $f(x) = -5x + 1$ .
2. We need to find the inverse function, denoted as $f^{-1}(x)$ .
3. The function is linear, so its inverse will also be a linear function.
4. The inverse function $f^{-1}(x)$ will satisfy the condition $f(f^{-1}(x)) = x$ .

STEP 2

1. Set $y = f(x)$ .
2. Solve for $x$ in terms of $y$ .
3. Swap $x$ and $y$ to find $f^{-1}(x)$ .
4. Verify the inverse function by checking the condition $f(f^{-1}(x)) = x$ .

STEP 3

Set $y = f(x)$ , which gives us:
$y = -5x + 1$

STEP 4

Solve for $x$ in terms of $y$ . Start by isolating the term with $x$ :
$y - 1 = -5x$
Divide both sides by $-5$ to solve for $x$ :
$x = \frac{y - 1}{-5}$

STEP 5

Swap $x$ and $y$ to express the inverse function:
$y = \frac{x - 1}{-5}$
Thus, the inverse function is:
$f^{-1}(x) = \frac{x - 1}{-5}$

STEP 6

Verify the inverse function by checking if $f(f^{-1}(x)) = x$ .
Substitute $f^{-1}(x)$ into $f(x)$ :
$f\left(\frac{x - 1}{-5}\right) = -5\left(\frac{x - 1}{-5}\right) + 1$
Simplify the expression:
$= (x - 1) + 1$ $= x$
Since $f(f^{-1}(x)) = x$ , the inverse function is verified.
The inverse function is:
$\boxed{\frac{x - 1}{-5}}$

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QuestionIf $f(x)=-5 x+1$ , then $f^{-1}(x)=$ $\square$
Submit Question

Studdy Solution

STEP 1

1. We are given the function $f(x) = -5x + 1$ .
2. We need to find the inverse function, denoted as $f^{-1}(x)$ .
3. The function is linear, so its inverse will also be a linear function.
4. The inverse function $f^{-1}(x)$ will satisfy the condition $f(f^{-1}(x)) = x$ .

STEP 2

1. Set $y = f(x)$ .
2. Solve for $x$ in terms of $y$ .
3. Swap $x$ and $y$ to find $f^{-1}(x)$ .
4. Verify the inverse function by checking the condition $f(f^{-1}(x)) = x$ .

STEP 3

Set $y = f(x)$ , which gives us:
$y = -5x + 1$

STEP 4

Solve for $x$ in terms of $y$ . Start by isolating the term with $x$ :
$y - 1 = -5x$
Divide both sides by $-5$ to solve for $x$ :
$x = \frac{y - 1}{-5}$

STEP 5

Swap $x$ and $y$ to express the inverse function:
$y = \frac{x - 1}{-5}$
Thus, the inverse function is:
$f^{-1}(x) = \frac{x - 1}{-5}$

STEP 6

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QuestionIf f(x)=−5x+1f(x)=-5 x+1f(x)=−5x+1, then f−1(x)=f^{-1}(x)=f−1(x)= □\square□Submit Question

Studdy Solution

STEP 1

STEP 2

1. Set y=f(x) y = f(x) y=f(x).2. Solve for x x x in terms of y y y.3. Swap x x x and y y y to find f−1(x) f^{-1}(x) f−1(x).4. Verify the inverse function by checking the condition f(f−1(x))=x f(f^{-1}(x)) = x f(f−1(x))=x.

STEP 3

Set y=f(x) y = f(x) y=f(x), which gives us:y=−5x+1 y = -5x + 1 y=−5x+1

STEP 4

Solve for x x x in terms of y y y. Start by isolating the term with x x x:y−1=−5x y - 1 = -5x y−1=−5xDivide both sides by −5-5−5 to solve for x x x:x=y−1−5 x = \frac{y - 1}{-5} x=−5y−1​

STEP 5

Swap x x x and y y y to express the inverse function:y=x−1−5 y = \frac{x - 1}{-5} y=−5x−1​Thus, the inverse function is:f−1(x)=x−1−5 f^{-1}(x) = \frac{x - 1}{-5} f−1(x)=−5x−1​

STEP 6

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QuestionIf $f(x)=-5 x+1$ , then $f^{-1}(x)=$ $\square$
Submit Question

1. Set $y = f(x)$ .
2. Solve for $x$ in terms of $y$ .
3. Swap $x$ and $y$ to find $f^{-1}(x)$ .
4. Verify the inverse function by checking the condition $f(f^{-1}(x)) = x$ .

Set $y = f(x)$ , which gives us:
$y = -5x + 1$

Solve for $x$ in terms of $y$ . Start by isolating the term with $x$ :
$y - 1 = -5x$
Divide both sides by $-5$ to solve for $x$ :
$x = \frac{y - 1}{-5}$

Swap $x$ and $y$ to express the inverse function:
$y = \frac{x - 1}{-5}$
Thus, the inverse function is:
$f^{-1}(x) = \frac{x - 1}{-5}$