Inverse Functions
In exercises 19., 20., 21., and 22. for each function we have to find the inverse. Before we find the
inverse function we have to find the domain where f is a one to one and non-decreasing.
19. First I graphed the original function and I made sure it was a one to one non-decreasing function.
Then I found the domain[-7,infinity]. Next I found the inverse of the function by switching the x and the y.
y=(x+7)^2
x=(y+7)^2
Sqrt[x]=y+7
y=Sqrt[x]-7
Plot[f (x) = (x + 7) ^ 2, {x, - 10, 10}]
Plot[f (x) = Sqrt[x] - 7, {x, - 10, 10}]
300
250
200
150
Out[309]=
100
50
-10
-5
5
10
5
10
-4.0
-4.5
-5.0
Out[310]=
-5.5
-6.0
-6.5
-10
-5
20. First I graphed the original function and I made sure it was a one to one non-decreasing function.
Then I found the domain[6,infinity]. Next I found the inverse of the function by switching the x and the y.
y=(x-6)^2
x=(y-6)^2
Printed by Wolfram Mathematica Student Edition
2
Sqrt[x]=y-6
y=Sqrt[x]+6
Plot[f (x) = (x - 6) ^ 2, {x, - 10, 10}]
Plot[f (x) = Sqrt[x] + 6, {x, - 10, 10}]
250
200
150
Out[311]=
100
50
-10
-5
5
10
5
10
9.0
8.5
8.0
Out[312]=
7.5
7.0
6.5
-10
-5
21. First I graphed the original function and I made sure it was a one to one non-decreasing function.
Then I found the domain[0,infinity]. Next I found the inverse of the function by switching the x and the y.
y=x^2-5
x=y^2-5
y^2=x+5
y=Sqrt(x+5)
Printed by Wolfram Mathematica Student Edition
3
In[313]:=
Plot[f (x) = x ^ 2 - 5, {x, - 10, 10}]
Plot[f (x) = Sqrt[x - 5], {x, - 10, 10}]
100
80
60
Out[313]=
40
20
-10
-5
5
10
5
10
2.0
1.5
Out[314]=
1.0
0.5
-10
-5
22. First I graphed the original function and I made sure it was a one to one non-decreasing function.
Then I found the domain[0,infinity]. Next I found the inverse of the function by switching the x and the y.
y=x^2+1
x=y^2+1
y^2=x-1
y=Sqrt(x-1)
Printed by Wolfram Mathematica Student Edition
4
In[315]:=
Plot[f (x) = x ^ 2 + 1, {x, - 10, 10}]
Plot[f (x) = Sqrt[x - 1], {x, - 10, 10}]
100
80
60
Out[315]=
40
20
-10
-5
5
10
5
10
3.0
2.5
2.0
Out[316]=
1.5
1.0
0.5
-10
-5
1. The difference between Sqrt[x]^2 and Sqrt[x^2] is that in Mathematica if you were to put Sqrt[x]^2 the
Sqrt and the ^2 would cancel each other out

Sqrt[x]^2
Sqrt[x] ^ 2
Out[317]=
x
If you were to put Sqrt[x^2] this would come out
In[318]:=

Sqrt[x ^ 2]
Out[318]=
x2
2. Plot x^2, Sqrt[x], x
Printed by Wolfram Mathematica Student Edition
5
In[353]:=
f[x_] := x ^ 2
g[x_] := Sqrt[x]
h[x_] := x
p1 = Plot[f[x], {x, - 10, 10}, PlotStyle → Blue]
p2 = Plot[g[x], {x, - 10, 10}, PlotStyle → Red]
p3 = Plot[h[x], {x, - 10, 10}, PlotStyle → Dashed]
Show[{p1, p2, p3}]
100
80
60
Out[356]=
40
20
-10
-5
5
10
To convert Fahrenheit to Celsius we have to use the inverse function of f[x_] := (5/9)*(x – 32) and it
would be g[x_]:=(9/5)×(x+32)
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