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10 Apr 2020
Discuss: Solving an Equation for an Unknown Function. In Exercise
of section
youu were asked to solve equations in which the unknowns are functions. Now that we know about inverse and the identity function (see exercise
), we can use algebra to solve
for unknown function
, we perform the following steps:
Problem: solve for ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAAMBAMAAACtsOGuAAAACXBIWXMAAABkAAAAZAAPlsXdAAAAJ1BMVEVHcEwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAB+jSoGAAAADHRSTlMAiHdE3VsimaoRzDNoTfY4AAAANElEQVQI12NgYNguxMDAAMIMzUA88WAAkHQAYuYGIMEkACTYDYAEVwKQYAHilEAgMQmoCgC+TQYSkuh7FQAAAABJRU5ErkJggg==)
Compose with
on the right
Because ![](data:image/png;base64,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)
Because ![](data:image/png;base64,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)
So the equation is
. Use this technique to solve the equation
for the indicated unknown function.
Solve for
, where
and ![](data:image/png;base64,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)
Discuss: Solving an Equation for an Unknown Function. In Exercise of section
youu were asked to solve equations in which the unknowns are functions. Now that we know about inverse and the identity function (see exercise
), we can use algebra to solve
for unknown function
, we perform the following steps:
Problem: solve for
Compose with
on the right
Because
Because
So the equation is . Use this technique to solve the equation
for the indicated unknown function.
Solve for , where
and