Let f(x) = ax + b, a linear function
To find inverse of the linear function f(x), we have to follow the steps given below.
Step 1 :
Say f(x) = y
Step 2 :
y = ax + b
The given function is derived for y, now we have to solve for the function to derive for x.
(y - b) = ax
x = (y - b) / a
Step 3 :
Change x as f^{-1}(x) and y as x.
f^{-1}(x) = (x - b) / a
Problem 1 :
For g(t) = 3t - 2, determine each value.
a) g(13) b) g(7)
c) [g(13) - g(7)] / ( 13 - 7)
d) g^{-1}(7)
e) g^{-1}(13)
f) [g^{-1}(13) - g^{-1}(7)] / (13 - 7)
Solution :
g(t) = 3t - 2
a) g(13) :
Applying t = 13, we get
g(13) = 3(13) - 2
= 39 - 2
= 37
b) g(7)
Applying t = 7, we get
g(7) = 3(7) - 2
= 21 - 2
= 19
c) [g(13) - g(7)] / ( 13 - 7)
g(13) = 37, g(7) = 19
Applying the values, we get
= (37 - 19) / (13 - 7)
= 18 / 6
= 3
d) g^{-1}(7)
Finding inverse function of the given function,
g(t) = 3t - 2
y = 3t - 2
Solving for t,
y + 2 = 3t
t = (y + 2) / 3
Replace t as g^{-1}(t) and y as t, we get
g^{-1}(t) = (t + 2) / 3
g^{-1}(7) = (7 + 2) / 3
= 9/3
g^{-1}(7) = 3
e) g^{-1}(13)
g^{-1}(t) = (t + 2) / 3
g^{-1}(13) = (13 + 2) / 3
= 15/3
g^{-1}(13) = 5
f) [g^{-1}(13) - g^{-1}(7)] / (13 - 7)
Applying the values above,
= (5 - 3)/(13 - 7)
= 2 / 6
= 1/3
Problem 2 :
The formula for converting a temperature in degrees Celsius into degrees Fahrenheit is
F = (9/5) C + 32
Shirelle, an American visitor to Canada, uses a simpler rule to convert from Celsius to Fahrenheit: Double the Celsius temperature, then add 30.
a) Use function notation to write an equation for this rule. Call the function f and let x represent the temperature in degrees Celsius.
b) Write f^{-1}(x)
c) One day, the temperature was 14 degree Celsius. Use function notation to express this temperature in degrees Fahrenheit.
d) Another day, the temperature was 70 degree Fahrenheit. Use function notation to express this temperature in degrees Celsius.
Solution :
a) x represents the temperature in degrees Celsius
f(x) = 2x + 30
b) Let y = f(x)
y = 2x + 30
y - 30 = 2x
(y - 30) / 2 = x
f^{-1}(x) = (x - 30)/2
c) F = (9/5) C + 32
F = (9/5) (14) + 32
= 126/5 + 32
= 25.2 + 32
F = 57.2
d) F = (9/5) C + 32
70 = (9/5) C + 32
70 - 32 = (9/5)C
38 = (9/5)C
C = (38 x 5) / 9
C = 21.1
Problem 3 :
Ben, another American visitor to Canada, uses this rule to convert centimeters to inches: Multiply by 4 and then divide by 10. Let the function g be the method for converting centimeters to inches, according to Ben’s rule.
a) Write g^{-1} (x) as a rule.
b) Describe a situation in which the rule g^{-1} (x) for might be useful
c) Determine g(x) and g^{-1} (x).
d) One day 15 cm of snow fell. Use function notation to represent this amount in inches.
e) Ben is 5 ft 10 in tall. Use function notation to represent the height in centimeters.
Solution :
Multiply by 4 and then divide by 10.
a) g(x) = 4x / 10
Let y = g(x)
y = 4x / 10
Solving for x, we get
10y = 4x
x = 10y/4
g^{-1}(x) = 10x / 4
b) Multiply by 10 and divide by 4.
c) g^{-1}(x) = 10x / 4
d) g(x) is the function to be used to convert between centimeters to inches.
g^{-1}(x) is the function to be used to convert between inches to centimeters.
g(x) = 4x / 10
When x = 15
g(15) = 4(15) / 10
= 60 / 10
= 6 inches
e) Ben is 5 ft 10 in tall. Use function notation to represent the height in centimeters.
1 ft = 12 inches
5 ft = 5 (12) ==> 60 inches
= 60 + 10
= 70 inches
g^{-1}(x) = 10x / 4
When x = 70
g^{-1}(70) = 10(70) / 4
= 175 cm
Problem 4 :
If f(x) = k(2 + x), find the value of k when f^{-1}(-2) = -3
Solution :
f(x) = k(2 + x)
Let y = f(x)
y = k(2 + x)
y = 2k + 2x
2x = y - 2k
x = (y - 2k) / 2
f^{-1}(x) = (x - 2k)/2
f^{-1}(-2) = -3
-3 = (-2 - 2k)/2
-6 = -2 - 2k
-6 + 2 = -2k
-2k = -4
k = 2
So, the value of k is 2.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM