Algebra 2
Graphing Inverses
Name ________________________
Review: Find the inverse of each equation:
1
1) y  x  12
2) y  8 x3
4
3) y  3 x  1
4.)Which of the following are one-to-one functions (there may be more than 1)? _____________
x7
y
A.) y  x  3
B.)
3
y
y
4
4
2
2
–4
–2
2
–4
x
4
–2
2
–2
–2
–4
–4
C.)
x
4
D.)
Graphing Inverses To graph an inverse, find points from the original graph then switch the x
and y coordinates. The inverse graph should be reflected over the y = x line. The Domain and
the Range will also be switched.
Draw the inverse of each given graph on the same plane. Then identify the Domain/Range for
the original and its inverse.
5.
6.
7.
y
–4
y
y
4
4
4
2
2
2
–2
2
4
x
–4
–2
2
4
x
–4
–2
2
–2
–2
–2
–4
–4
–4
4
Domain of f(x)__________
Range of f(x) ___________
Domain of f(x)__________
Range of f(x) ___________
Domain of f(x)__________
Range of f(x) ___________
Domain of f-1 (x) _________
Range of f-1 (x) _________
Domain of f-1 (x) _________
Range of f-1 (x) __________
Domain of f-1 (x) _________
Range of f-1 (x) _________
x
Inverses of Exponential/Logarithms :
The inverse of an exponential equation is a_______________. When we find the inverse of an
exponential equation, we still switch the x and y as previously, but when we solve for y we actually
rewrite the equation in a different form using the word ‘log’.
Example 1: Find the inverse of: y  2 x  Switching x and y gives us: x  2 y
To solve for y, we simply write log 2 x  y
1.) Graph y  2 x . Use a t-table to plot points. Don’t forget the
asymptote!
y
4
2
–4
–2
2
–2
–4
4
x
2.) Graph the inverse: y  log 2 x on the same plane by switching
the coordinates. Did the asymptote change?
3.) Like other functions, the graphs of the original equation and its
inverse are reflected over the line _____________.Sketch this line.
4.) Give the Domain/Range of the exponential and the log functions graphed above:
Domain of expon: ____________
Range of expon.: ______________
Domain of Log: ______________
Range of Log: ________________
Converting Exponential/Logarithm equations: It is sometimes helpful when we solve exponential and
log equations if we can convert the equations back and forth. (We will be solving later in the semester.)
Converting these equations from one form to another uses similar steps to finding the inverse.
Example 2: Convert from exponential to log form: 23  8  log 2 8  3
Notice: the word ‘log’ is written 1st, the base of 2 becomes the little base of 2 in the log
equation and the exponent of 3 is on the other side of the = sign.
It is important to understand that:
Logarithms are ______________________!
Convert the following to log form:
4.) 3x  27
5.) x 2  3
6.) 54  x
7.) 5 x 2  125
Work backwards to convert the following to exponential form: (Write the base 1st and remember, the
exponent will be on the other side of = )
8.) log x 25  2
9.) log3 ( x  2)  4
10.) log 7 49  x
11.) log x3 165  2
Algebra 2
Name __________________________
Homework-Graphing inverses & converting log/expon
Draw the inverse of each given graph on the same plane. Draw the line y = x. Then identify the
Domain/Range for the original and its inverse.
1.
2.
3.
y
–4
y
4
4
4
2
2
2
–2
2
x
4
–4
–2
What is the
name of this
graph?
y
2
4
x
–4
–2
2
4
–2
–2
–2
–4
–4
–4
Domain of f(x)__________
Range of f(x) ___________
Domain of f(x)__________
Range of f(x) ___________
Domain of f(x)__________
Range of f(x) ___________
Domain of f-1 (x) _________
Range of f-1 (x) _________
Domain of f-1 (x) _________
Range of f-1 (x) __________
Domain of f-1 (x) _________
Range of f-1 (x) _________
x
What is the
name of its
inverse?
(Thinking
about its
shape may
help you in
sketching the
inverse.)
Convert each equation from log to exponential or from exponential to log form:
5. 3x  27
6. log b 9  2
7. 5 y  x
8. 25 2  x
9. log 1 10 x  3
10. log b 0.01  2
11. 85  3x  1
12. x 2  3
13. log 7 ( x 2  16)  80
4. log 2 x  5
3
14. log x  2 16  2
15. 21  3 x  1
16. The graph for y  log3 x is shown below. Graph its inverse and the new asymptote.
y
8
17. What is the domain for y  log3 x ? The range?
6
4
2
–8
–6
–4
–2
2
4
6
8
x
18. What is the equation of the inverse?
Is it a function?
–2
–4
–6
–8
19. What is the domain of the inverse? The range?