Interchange the variables.

Rewrite the equation as <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⋅</mo><mi>sin</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi></mstyle></math> .

Multiply each term by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and simplify.

Multiply each term in <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⋅</mo><mi>sin</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Simplify <math><mstyle displaystyle="true"><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⋅</mo><mi>sin</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>⋅</mo><mn>2</mn></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Move <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> to the left of <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Take the inverse sine of both sides of the equation to extract <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> from inside the sine.

Replace the <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> with <math><mstyle displaystyle="true"><msup><mrow><mi>f</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> to show the final answer.

Set up the composite result function.

Evaluate <math><mstyle displaystyle="true"><mi>g</mi><mrow><mo>(</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow></mstyle></math> by substituting in the value of <math><mstyle displaystyle="true"><mi>f</mi></mstyle></math> into <math><mstyle displaystyle="true"><mi>g</mi></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Since <math><mstyle displaystyle="true"><mi>g</mi><mrow><mo>(</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mi>x</mi></mstyle></math> , <math><mstyle displaystyle="true"><msup><mrow><mi>f</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>arcsin</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> is the inverse of <math><mstyle displaystyle="true"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⋅</mo><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

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