Functions

Functions

Inverse and composite functions

Learning Objectives

• Understand about inverse of a function

• Determine of inverse of a functions

• Form composite functions

Multiple Choice

Which of the following relation represents a function?

• {(1,2), (1,3), (2,3)}

• {(1,2), (1,3), (1,4)}

• {(1,2), (2,3), (2,3)}

• {(1,2), (2,3), (3,3)}

Multiple Choice

Is this graph a function or not a function?

• Function

• Not a Function

Multiple Choice

Is this mapping a function or not a function? 

• Function
• Not a Function

Multiple Choice

For the function {(0,1), (1,-3), (2,-4), (-4,1)}, write the domain and range.

• D: {1, -3, -4,}
  R: {0, 1, 2, -4}
• D:{0, 1, 2, -4}
  R:{1, -3, -4}
• D:{0, 1, 2, 3, 4}
  R:{1, -3, -4}

​Review

• If input is 4, output is 3, OR f(4) = 3

• If output is 3, input is 4 OR f^-1(3) = 4

  .

• What is f(1), f^-1(1)?

• What is f^-1(x)?

​

Multiple Choice

What are the missing values if g(x)=f^-1(x)?

• (9,-5)
• (-9,-5)
• (5,9)
• (-9,5)

Multiple Choice

Given: $$f\left(x\right)=2x-3$$  .

What is $$f^{-1}\left(1\right)?$$  

• $$-1$$  

• 1

• 2

• 8

Multiple Choice

Determine f^-1(2) given the following information about an invertible function, f(x)

f(1) = -3

f(2) = -2

f(-1) = 2

f(3) = -1

• 1

• -1

• 2

• -2

Do all functions have inverse?

If a function gives the same value of y for two or more values of x, it does not have an inverse function.

Functions with inverse

If a horizontal line cuts the graph of a function at most once, then that function has an inverse.

Functions with NO inverse

If a horizontal line cuts the graph more than once, then that function does not have an inverse.

Multiple Choice

Which graph passes the horizontal line test?

• Graph 1

• Graph 2

• Graph 3

• Graph 4

Multiple Choice

Does this graph have an inverse?

• Yes

• No

Multiple Choice

Does this graph have an inverse?

• Yes

• No

Multiple Choice

f(x) = 3x + 10
g(x) = x - 2
Find f(g(5))

• 19
• 23
• -10
• None of these.

Multiple Choice

f(x) = 3x + 10

g(x) = x - 2

Find f(g(0))

• 16

• 4

• -4

• None of these.

Multiple Choice

If f(x) = 3x-1 and g(x) = x²+2,

what is (f ° g)(x) ?

• 3x² +5

• x² +1

• 3x² +1

• 3x² +6

Multiple Choice

If $$f\left(x\right)=x^2$$   and  $$g\left(x\right)=\frac{1}{x^2}$$   find $$\left(f\circ g\right)\left(-1\right)$$  

• $$\frac{1}{4}$$  

• $$1$$

• $$-1$$  

• $$-\frac{1}{4}$$  

Multiple Choice

If $$f\left(x\right)=\frac{1}{x}$$   and  $$g\left(x\right)=3x+2$$   find $$\left(g\circ g\right)\left(-4\right)$$  

• 44

• -40

• -30

• -28

Multiple Choice

If $$f\left(x\right)=\frac{1}{x}$$   and  $$g\left(x\right)=3x+2$$   find $$\left(f\circ g\right)\left(2\right)$$  

• $$\frac{1}{4}$$  

• $$\frac{1}{8}$$  

• $$\frac{7}{2}$$  

• 8

Multiple Choice

If $$f\left(x\right)=2x$$   and  $$g\left(x\right)=2x^2-1$$   find $$g\left(f\left(x\right)\right)$$  

• $$4x^2-1$$  

• $$4x^2-2$$  

• $$8x^2-1$$  

• $$16x^2-1$$  

Multiple Choice

If $$f\left(x\right)=2x$$   and  $$g\left(x\right)=2x^2-1$$   find $$f\left(g\left(x\right)\right)$$  

• $$4x^2-1$$  

• $$4x^2-2$$  

• $$8x^2-1$$  

• $$16x^2-1$$  

Multiple Choice

If $$f\left(x\right)=2x$$   and  $$g\left(x\right)=2x^2-1$$   find $$g\left(f\left(-1\right)\right)$$  

• 2

• 15

• 7

• -9

Multiple Choice

If $$f\left(x\right)=2x$$   and  $$g\left(x\right)=2x^2-1$$   find $$f\left(g\left(3\right)\right)$$  

• 34

• 71

• 35

• 142