How to find inverse of a algebraic equation of cubic function?
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How do I find the inverse of a cubic function?
How can I double-check my answer on paper and with a graphing calculator?
When I need to write an inverse function, I rewrote the equation with opposite terms, then solve it, but not all functions have inverses.
First: Is the function one-to-one?
Second: If the equation starts with f(x), change f(x) to y
Third: Reverse the terms. Change y to x, change each x to y.
Fourth: Solve the new equation for y
Fifth: Change the y to f(x).
Let me illustrate with f(x) = x³ + 3
First: Is the function one-to-one?
Most Algebra students can tell you that this is invertible. One way to be sure if it one-to-one is to look at the graph. Here is the graph on my Texas Instruments TI-84 PLUS CE Graphing Calculator

There is no horizontal line that would go through two points, so an inverse function is, indeed, possible for:
→ f(x) = x³ + 3
Second: If the equation starts with f(x), change f(x) to y
→ y = x³ + 3
Third: Reverse the terms. Change y to x, change each x to y.
→ x = y³ + 3
Fourth: Solve the new equation for y
→ y³ = x - 3
→ y = (x - 3)1/31/3
→ y = x−3−−−−−√3x−33
Fifth: Change the y to f(x).
→ f(x) = x−3−−−−−√3x−33
How can I double-check my answer on paper?
Sketch both functions on the same graph. Are the two graphs symmetrical across the diagonal (f(x) = x) line? If you put a mirror on the (y=x) line, will you see the same graph? (See graph below.)
How can I double-check my answer with a graphing calculator?

As you can see in this example, I graphed four functions:
Y1=X3+3Y1=X3+3Y2=(X−3)1/3Y2=(X−3)1/3Y3=XY3=XY4=(Y1–3)1/3Y4=(Y1–3)1/3
Equation 1 shows the equation I started with. Equation 2 shows the answer we found together. Equation 3 shows the diagonal line.
With these three equations, we can use our eyes and ask ourselves. is this a symmetrical equation? Yes, I can look at this and see that it is just fine.
BUT … I like to be sure, so I type one more equation. Look at equation 4. I have taken my answer equation:
Y2=(X−3)1/3Y2=(X−3)1/3
and rewritten it in terms of Y1Y1
Y4=(Y1–3)1/3Y4=(Y1–3)1/3
If the answer is correct, you will be able to watch your calculator draw the third equation in one color, then immediately redraw it in the next color. As you can see in the image above, the diagonal line is drawn in green, but before that line was drawn, I watched it draw the same line in purple.
I used this double-check method to help me get 100% in College Algebra (the third course offered at a Tech College) this summer. I skipped the intro and intermediate course because I assumed that I remembered enough from fifty years ago. (Part of my goal in taking the course again was to learn to use the graphing calculator). On several occasions when doing homework and quizzes for that course, I did not get a diagonal line, so that let me go back to my paper and figure out my error.
How can I double-check my answer on paper and with a graphing calculator?
When I need to write an inverse function, I rewrote the equation with opposite terms, then solve it, but not all functions have inverses.
First: Is the function one-to-one?
Second: If the equation starts with f(x), change f(x) to y
Third: Reverse the terms. Change y to x, change each x to y.
Fourth: Solve the new equation for y
Fifth: Change the y to f(x).
Let me illustrate with f(x) = x³ + 3
First: Is the function one-to-one?
Most Algebra students can tell you that this is invertible. One way to be sure if it one-to-one is to look at the graph. Here is the graph on my Texas Instruments TI-84 PLUS CE Graphing Calculator

There is no horizontal line that would go through two points, so an inverse function is, indeed, possible for:
→ f(x) = x³ + 3
Second: If the equation starts with f(x), change f(x) to y
→ y = x³ + 3
Third: Reverse the terms. Change y to x, change each x to y.
→ x = y³ + 3
Fourth: Solve the new equation for y
→ y³ = x - 3
→ y = (x - 3)1/31/3
→ y = x−3−−−−−√3x−33
Fifth: Change the y to f(x).
→ f(x) = x−3−−−−−√3x−33
How can I double-check my answer on paper?
Sketch both functions on the same graph. Are the two graphs symmetrical across the diagonal (f(x) = x) line? If you put a mirror on the (y=x) line, will you see the same graph? (See graph below.)
How can I double-check my answer with a graphing calculator?

As you can see in this example, I graphed four functions:
Y1=X3+3Y1=X3+3Y2=(X−3)1/3Y2=(X−3)1/3Y3=XY3=XY4=(Y1–3)1/3Y4=(Y1–3)1/3
Equation 1 shows the equation I started with. Equation 2 shows the answer we found together. Equation 3 shows the diagonal line.
With these three equations, we can use our eyes and ask ourselves. is this a symmetrical equation? Yes, I can look at this and see that it is just fine.
BUT … I like to be sure, so I type one more equation. Look at equation 4. I have taken my answer equation:
Y2=(X−3)1/3Y2=(X−3)1/3
and rewritten it in terms of Y1Y1
Y4=(Y1–3)1/3Y4=(Y1–3)1/3
If the answer is correct, you will be able to watch your calculator draw the third equation in one color, then immediately redraw it in the next color. As you can see in the image above, the diagonal line is drawn in green, but before that line was drawn, I watched it draw the same line in purple.
I used this double-check method to help me get 100% in College Algebra (the third course offered at a Tech College) this summer. I skipped the intro and intermediate course because I assumed that I remembered enough from fifty years ago. (Part of my goal in taking the course again was to learn to use the graphing calculator). On several occasions when doing homework and quizzes for that course, I did not get a diagonal line, so that let me go back to my paper and figure out my error.
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