Functions and Their Inverses - Humble Independent … · to verify that 2 functions are inverses....
Transcript of Functions and Their Inverses - Humble Independent … · to verify that 2 functions are inverses....
Holt Algebra 2
9-5 Functions and Their Inverses
To find the inverse of a function
• Switch the x and y
• Solve for y
Holt Algebra 2
9-5 Functions and Their Inverses
Example 1 𝑔(𝑥) =3
4𝑥
𝑥 =3
4𝑦
4
3𝑥 =
3
4𝑦
4
3
4
3𝑥 = 𝑦
𝑔−1 𝑥 =4
3𝑥
𝑔−1 𝑥
Means the inverse of g(x)
Holt Algebra 2
9-5 Functions and Their Inverses
Example 2
𝑔−1 𝑥
Means the inverse of g(x)
𝑔 𝑥 = 9𝑥 − 5
𝑥 = 9𝑦 − 5
𝑥 + 5 = 9𝑦
𝑥 + 5
9=9𝑦
9
𝑥 + 5
9= 𝑦
𝑔−1(𝑥) =𝑥 + 5
9
Holt Algebra 2
9-5 Functions and Their Inverses
Example 2 Continued
Rewrite the function using y instead of f(x).
Step 1 Find the inverse.
Simplify.
Switch x and y in the equation.
Cube both sides.
Isolate y.
Holt Algebra 2
9-5 Functions and Their Inverses
Rewrite the function using y instead of f(x).
Step 1 Find the inverse.
Take the cube root of both
sides.
Switch x and y in the equation.
Add 2 to both sides of the equation.
Simplify.
Check It Out! Example 2 Continued
y = x3
– 2
x = y3
– 2
x + 2 = y3
3 x + 2 = y
33 3x + 2 = y
Holt Algebra 2
9-5 Functions and Their Inverses
The inverse functions “undo” each other,
You can use composition of functions
to verify that 2 functions are inverses.
When you compose two inverses…
the result is the input value of x.
Holt Algebra 2
9-5 Functions and Their Inverses
If f(g(x)) = g(f(x)) = x
Then f(x) and g(x) are inverse functions
Example 1:
f g x g f x x
3f g x x 1
3g f x x
Because f(g(x)) = g(f(x)) = x, they are inverses.
Holt Algebra 2
9-5 Functions and Their Inverses
Determine by composition whether each pair of functions are inverses.
Example 2:
Find the composition f(g(x)).
Use the Distributive
Property.
Simplify.
f(x) = 3x – 1 and g(x) = x + 1 1
3
Substitute x + 1 for
x in f.
1
3
= (x + 3) – 1
f(g(x)) = x + 2
The functions are NOT inverses.
3 1f g x x
31 1
1 13 3
1x xf
f g x
Holt Algebra 2
9-5 Functions and Their Inverses
Example 3 Determine by composition whether each pair of functions are inverses.
Find the composition f(g(x)) and g(f(x)).
3
2f(x) = x + 6 and g(x) = x – 9 2
3
= x – 6 + 6 = x + 9 – 9
Because f(g(x)) = g(f(x)) = x, they are inverses.
2
63
f g x x
26
3
3 39 9
2 2xf x
f g x x g f x x
3
92
g f x x
39
2
2 26 6
3 3xg x
Holt Algebra 2
9-5 Functions and Their Inverses
Find the compositions f(g(x)) and g(f(x)).
Simplify.
Example 4
Substitute for x
in f.
f(x) = x2 + 5 and for x ≥ 0
= x + 25 +5 10 x
Because f(g(x)) ≠ x,f (x) and g(x) are NOT inverses.
2( ( )) 5f g x x
2
5 55x xf
10 30f g x x x
Holt Algebra 2
9-5 Functions and Their Inverses
Example 5: Are the following functions Inverses?
For x ≠ 1 or 0, f(x) = and g(x) = + 1. 1
x
1
x – 1
Because f(g(x)) = g(f (x)) = x
f(x) and g(x) are inverses.
= (x – 1) + 1
gf x x
1( ( ))
1f g x
x
1 1
11
1
1x
fx
1
x
1
x
1( ( )) 1g f x
x
11
11
1
1x
x
g
1
1
x
1
1
x
( )g f x x
Holt Algebra 2
9-5 Functions and Their Inverses
Lesson Quiz: Part I
A: yes; B: no
1. Use the horizontal-line test to determine whether the inverse of each relation is a function.
Holt Algebra 2
9-5 Functions and Their Inverses
Lesson Quiz: Part II
D: {x|x ≥ 4}; R: {all Real Numbers}
2. Find the inverse f(x) = x2
– 4. Determine whether it is a function, and state its domain and range.
not a function