Find the Inverse d=0.2t^(3/2)

$d = 0.2 t^{\frac{3}{2}}$

Step 1

Interchange the variables.

$t = 0.2 y^{\frac{3}{2}}$

Step 2

Solve for $y$ .

Rewrite the equation as $0.2 y^{\frac{3}{2}} = t$ .

$0.2 y^{\frac{3}{2}} = t$

Divide each term in $0.2 y^{\frac{3}{2}} = t$ by $0.2$ and simplify.

Divide each term in $0.2 y^{\frac{3}{2}} = t$ by $0.2$ .

$\frac{0.2 y^{\frac{3}{2}}}{0.2} = \frac{t}{0.2}$

Simplify the left side.

Cancel the common factor.

$\frac{0.2 y^{\frac{3}{2}}}{0.2} = \frac{t}{0.2}$

Divide $y^{\frac{3}{2}}$ by $1$ .

$y^{\frac{3}{2}} = \frac{t}{0.2}$

Simplify the right side.

Multiply by $1$ .

$y^{\frac{3}{2}} = \frac{1 t}{0.2}$

Factor $0.2$ out of $0.2$ .

$y^{\frac{3}{2}} = \frac{1 t}{0.2 (1)}$

Separate fractions.

$y^{\frac{3}{2}} = \frac{1}{0.2} \cdot \frac{t}{1}$

Divide $1$ by $0.2$ .

$y^{\frac{3}{2}} = 5 \frac{t}{1}$

Divide $t$ by $1$ .

$y^{\frac{3}{2}} = 5 t$

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{3}{2}})}^{\frac{2}{3}} = {(5 t)}^{\frac{2}{3}}$

Simplify the exponent.

Simplify the left side.

Simplify ${(y^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Multiply the exponents in ${(y^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{\frac{3}{2} \cdot \frac{2}{3}} = {(5 t)}^{\frac{2}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y^{\frac{3}{2} \cdot \frac{2}{3}} = {(5 t)}^{\frac{2}{3}}$

Rewrite the expression.

$y^{\frac{1}{2} \cdot 2} = {(5 t)}^{\frac{2}{3}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {(5 t)}^{\frac{2}{3}}$

Rewrite the expression.

$y^{1} = {(5 t)}^{\frac{2}{3}}$

Simplify.

$y = {(5 t)}^{\frac{2}{3}}$

Simplify the right side.

Apply the product rule to $5 t$ .

$y = 5^{\frac{2}{3}} t^{\frac{2}{3}}$

Step 3

Replace $y$ with $f^{- 1} (t)$ to show the final answer.

$f^{- 1} (t) = 5^{\frac{2}{3}} t^{\frac{2}{3}}$

Step 4

Verify if $f^{- 1} (t) = 5^{\frac{2}{3}} t^{\frac{2}{3}}$ is the inverse of $f (t) = 0.2 t^{\frac{3}{2}}$ .

To verify the inverse, check if $f^{- 1} (f (t)) = t$ and $f (f^{- 1} (t)) = t$ .

Evaluate $f^{- 1} (f (t))$ .

Set up the composite result function.

$f^{- 1} (f (t))$

Evaluate $f^{- 1} (0.2 t^{\frac{3}{2}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} {(0.2 t^{\frac{3}{2}})}^{\frac{2}{3}}$

Apply the product rule to $0.2 t^{\frac{3}{2}}$ .

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} ({0.2}^{\frac{2}{3}} {(t^{\frac{3}{2}})}^{\frac{2}{3}})$

Raise $0.2$ to the power of $\frac{2}{3}$ .

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} (0.34199518 {(t^{\frac{3}{2}})}^{\frac{2}{3}})$

Multiply the exponents in ${(t^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} (0.34199518 t^{\frac{3}{2} \cdot \frac{2}{3}})$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} (0.34199518 t^{\frac{3}{2} \cdot \frac{2}{3}})$

Rewrite the expression.

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} (0.34199518 t^{\frac{1}{2} \cdot 2})$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} (0.34199518 t^{\frac{1}{2} \cdot 2})$

Rewrite the expression.

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} (0.34199518 t)$

Multiply $0.34199518$ by $5^{\frac{2}{3}}$ .

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 1 t$

Evaluate $f (f^{- 1} (t))$ .

Set up the composite result function.

$f (f^{- 1} (t))$

Evaluate $f (5^{\frac{2}{3}} t^{\frac{2}{3}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 {(5^{\frac{2}{3}} t^{\frac{2}{3}})}^{\frac{3}{2}}$

Apply the product rule to $5^{\frac{2}{3}} t^{\frac{2}{3}}$ .

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 ({(5^{\frac{2}{3}})}^{\frac{3}{2}} {(t^{\frac{2}{3}})}^{\frac{3}{2}})$

Multiply the exponents in ${(5^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5^{\frac{2}{3} \cdot \frac{3}{2}} {(t^{\frac{2}{3}})}^{\frac{3}{2}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5^{\frac{2}{3} \cdot \frac{3}{2}} {(t^{\frac{2}{3}})}^{\frac{3}{2}})$

Rewrite the expression.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5^{\frac{1}{3} \cdot 3} {(t^{\frac{2}{3}})}^{\frac{3}{2}})$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5^{\frac{1}{3} \cdot 3} {(t^{\frac{2}{3}})}^{\frac{3}{2}})$

Rewrite the expression.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5 {(t^{\frac{2}{3}})}^{\frac{3}{2}})$

Multiply the exponents in ${(t^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5 t^{\frac{2}{3} \cdot \frac{3}{2}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5 t^{\frac{2}{3} \cdot \frac{3}{2}})$

Rewrite the expression.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5 t^{\frac{1}{3} \cdot 3})$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5 t^{\frac{1}{3} \cdot 3})$

Rewrite the expression.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5 t)$

Multiply $0.2 (5 t)$ .

Multiply $5$ by $0.2$ .

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 1 t$

Multiply $t$ by $1$ .

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = t$

Since $f^{- 1} (f (t)) = t$ and $f (f^{- 1} (t)) = t$ , then $f^{- 1} (t) = 5^{\frac{2}{3}} t^{\frac{2}{3}}$ is the inverse of $f (t) = 0.2 t^{\frac{3}{2}}$ .

$f^{- 1} (t) = 5^{\frac{2}{3}} t^{\frac{2}{3}}$

Find the Inverse d=0.2t^(3/2)

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