Calculus Examples

Find the Inverse cos(x)
cos(x)cos(x)
Step 1
Interchange the variables.
x=cos(y)x=cos(y)
Step 2
Solve for yy.
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Step 2.1
Rewrite the equation as cos(y)=xcos(y)=x.
cos(y)=xcos(y)=x
Step 2.2
Take the inverse cosine of both sides of the equation to extract yy from inside the cosine.
y=arccos(x)y=arccos(x)
Step 2.3
Remove parentheses.
y=arccos(x)y=arccos(x)
y=arccos(x)y=arccos(x)
Step 3
Replace yy with f-1(x)f1(x) to show the final answer.
f-1(x)=arccos(x)f1(x)=arccos(x)
Step 4
Verify if f-1(x)=arccos(x)f1(x)=arccos(x) is the inverse of f(x)=cos(x)f(x)=cos(x).
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Step 4.1
To verify the inverse, check if f-1(f(x))=xf1(f(x))=x and f(f-1(x))=xf(f1(x))=x.
Step 4.2
Evaluate f-1(f(x))f1(f(x)).
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Step 4.2.1
Set up the composite result function.
f-1(f(x))f1(f(x))
Step 4.2.2
Evaluate f-1(cos(x))f1(cos(x)) by substituting in the value of ff into f-1f1.
f-1(cos(x))=arccos(cos(x))f1(cos(x))=arccos(cos(x))
f-1(cos(x))=arccos(cos(x))f1(cos(x))=arccos(cos(x))
Step 4.3
Evaluate f(f-1(x))f(f1(x)).
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Step 4.3.1
Set up the composite result function.
f(f-1(x))f(f1(x))
Step 4.3.2
Evaluate f(arccos(x))f(arccos(x)) by substituting in the value of f-1f1 into ff.
f(arccos(x))=cos(arccos(x))f(arccos(x))=cos(arccos(x))
Step 4.3.3
The functions cosine and arccosine are inverses.
f(arccos(x))=xf(arccos(x))=x
f(arccos(x))=xf(arccos(x))=x
Step 4.4
Since f-1(f(x))=xf1(f(x))=x and f(f-1(x))=xf(f1(x))=x, then f-1(x)=arccos(x)f1(x)=arccos(x) is the inverse of f(x)=cos(x)f(x)=cos(x).
f-1(x)=arccos(x)f1(x)=arccos(x)
f-1(x)=arccos(x)f1(x)=arccos(x)
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