QuestionFind the fixed point of the function where $f(x)=\frac{2}{3} x-\frac{1}{2}$ , i.e., solve $f(x)=x$ .

Studdy Solution

STEP 1

Assumptions1. The function is defined as $f(x)=\frac{}{3} x-\frac{1}{}$ . A fixed point of a function is a value of $x$ for which $f(x)=x$

STEP 2

To find the fixed point, we need to set the function equal to $x$ and solve for $x$ .
$f(x) = x$

STEP 3

Substitute the given function into the equation.
$\frac{2}{3} x-\frac{1}{2} = x$

STEP 4

To solve for $x$ , we need to isolate $x$ on one side of the equation. First, we can subtract $\frac{2}{3}x$ from both sides to get all terms involving $x$ on one side.
$-\frac{1}{2} = x - \frac{2}{3} x$

STEP 5

Next, we can simplify the right side of the equation by combining like terms.
$-\frac{1}{2} = \frac{1}{3} x$

STEP 6

Finally, to solve for $x$ , we can multiply both sides of the equation by3.
$x = -\frac{1}{2} \times3$

STEP 7

Calculate the value of $x$ .
$x = -\frac{3}{2}$ The fixed point of the function $f(x)=\frac{2}{3} x-\frac{1}{2}$ is $x = -\frac{3}{2}$ .

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QuestionFind the fixed point of the function where $f(x)=\frac{2}{3} x-\frac{1}{2}$ , i.e., solve $f(x)=x$ .

Studdy Solution

STEP 1

Assumptions1. The function is defined as $f(x)=\frac{}{3} x-\frac{1}{}$ . A fixed point of a function is a value of $x$ for which $f(x)=x$

STEP 2

To find the fixed point, we need to set the function equal to $x$ and solve for $x$ .
$f(x) = x$

STEP 3

Substitute the given function into the equation.
$\frac{2}{3} x-\frac{1}{2} = x$

STEP 4

To solve for $x$ , we need to isolate $x$ on one side of the equation. First, we can subtract $\frac{2}{3}x$ from both sides to get all terms involving $x$ on one side.
$-\frac{1}{2} = x - \frac{2}{3} x$

STEP 5

Next, we can simplify the right side of the equation by combining like terms.
$-\frac{1}{2} = \frac{1}{3} x$

STEP 6

Finally, to solve for $x$ , we can multiply both sides of the equation by3.
$x = -\frac{1}{2} \times3$

STEP 7

Calculate the value of $x$ .
$x = -\frac{3}{2}$ The fixed point of the function $f(x)=\frac{2}{3} x-\frac{1}{2}$ is $x = -\frac{3}{2}$ .

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QuestionFind the fixed point of the function where f(x)=23x−12f(x)=\frac{2}{3} x-\frac{1}{2}f(x)=32​x−21​, i.e., solve f(x)=xf(x)=xf(x)=x.

Studdy Solution

STEP 1

Assumptions1. The function is defined as f(x)=3x−1f(x)=\frac{}{3} x-\frac{1}{}f(x)=3​x−1​ . A fixed point of a function is a value of xxx for which f(x)=xf(x)=xf(x)=x

STEP 2

To find the fixed point, we need to set the function equal to xxx and solve for xxx.f(x)=xf(x) = xf(x)=x

STEP 3

Substitute the given function into the equation.23x−12=x\frac{2}{3} x-\frac{1}{2} = x32​x−21​=x

STEP 4

To solve for xxx, we need to isolate xxx on one side of the equation. First, we can subtract 23x\frac{2}{3}x32​x from both sides to get all terms involving xxx on one side.−12=x−23x-\frac{1}{2} = x - \frac{2}{3} x−21​=x−32​x

STEP 5

Next, we can simplify the right side of the equation by combining like terms.−12=13x-\frac{1}{2} = \frac{1}{3} x−21​=31​x

STEP 6

Finally, to solve for xxx, we can multiply both sides of the equation by3.x=−12×3x = -\frac{1}{2} \times3x=−21​×3

STEP 7

Calculate the value of xxx.x=−32x = -\frac{3}{2}x=−23​The fixed point of the function f(x)=23x−12f(x)=\frac{2}{3} x-\frac{1}{2}f(x)=32​x−21​ is x=−32x = -\frac{3}{2}x=−23​.

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QuestionFind the fixed point of the function where $f(x)=\frac{2}{3} x-\frac{1}{2}$ , i.e., solve $f(x)=x$ .

Assumptions1. The function is defined as $f(x)=\frac{}{3} x-\frac{1}{}$ . A fixed point of a function is a value of $x$ for which $f(x)=x$

To find the fixed point, we need to set the function equal to $x$ and solve for $x$ .
$f(x) = x$

Substitute the given function into the equation.
$\frac{2}{3} x-\frac{1}{2} = x$

To solve for $x$ , we need to isolate $x$ on one side of the equation. First, we can subtract $\frac{2}{3}x$ from both sides to get all terms involving $x$ on one side.
$-\frac{1}{2} = x - \frac{2}{3} x$

Next, we can simplify the right side of the equation by combining like terms.
$-\frac{1}{2} = \frac{1}{3} x$

Finally, to solve for $x$ , we can multiply both sides of the equation by3.
$x = -\frac{1}{2} \times3$

Calculate the value of $x$ .
$x = -\frac{3}{2}$ The fixed point of the function $f(x)=\frac{2}{3} x-\frac{1}{2}$ is $x = -\frac{3}{2}$ .