H2 Maths 1.1 - Quadratic Inverse by Restricting Domain

Step 1 - Set up the problem

We want the inverse of f of x equals two x squared minus three x minus one.

Step 1 - Set up the problem

But a full quadratic is not one-to-one, so it does not have an inverse on its whole domain.

Step 1 - Set up the problem

First choose a single branch.

Step 2 - Find the turning point and choose a branch

Differentiate to find the turning point.

Step 2 - Find the turning point and choose a branch

F prime of x equals four x minus three, so the stationary point is at x equals three quarters.

Step 2 - Find the turning point and choose a branch

For x greater than or equal to three quarters, the derivative is non-negative, so the function is increasing and one-to-one.

Step 2 - Find the turning point and choose a branch

That is the branch we will keep for the inverse.

Step 3A - Complete the square

With the branch fixed, rewrite the function in completed-square form.

Step 3A - Complete the square

First factor out the two from the quadratic terms.

Step 3A - Complete the square

Then complete the square inside the bracket.

Step 3B - Finish the square and read the range

Multiplying by two turns negative nine over sixteen into negative nine over eight, so the constant term becomes negative seventeen over eight.

Step 3B - Finish the square and read the range

That gives the minimum value of the restricted function.

Step 3B - Finish the square and read the range

So the range starts at negative seventeen over eight.

Step 4 - Solve for the inverse

Now set f of x equal to y, then solve for x.

Step 4 - Solve for the inverse

Because we chose x greater than or equal to three quarters, we keep the positive square-root branch.

Step 4 - Solve for the inverse

Simplify, then swap x and y to write the inverse function.

Step 5 - State the final answer with domain and pitfall

So the inverse is three plus the square root of 8x plus 17, all over 4.

Step 5 - State the final answer with domain and pitfall

Its domain is x greater than or equal to negative seventeen over eight, because that was the original range.

Step 5 - State the final answer with domain and pitfall

Its range is y greater than or equal to three quarters, because we kept the right branch.

Step 5 - State the final answer with domain and pitfall

Common pitfall: if you choose the left branch instead, the square root comes with a minus sign.