Holooly Plus Logo
Holooly Plus Logo

Question 2.8.9: Finding Functions That Form a Given Composite Find functions...

Finding Functions That Form a Given Composite

Find functions ƒ and g such that

(ƒ ∘ g) (x) = (x²- 5)³ – 4(x²- 5) + 3.

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

Note the repeated quantity x² – 5. If we choose g(x) = x² – 5 and ƒ(x) = x³ – 4x + 3, then we have the following.

(ƒ ∘ g)(x)

= ƒ(g(x))                                   By definition

= ƒ(x² – 5)                                 g(x) = x² – 5

= (x² – 5)³ – 4(x² – 5) + 3       Use the rule for ƒ.

There are other pairs of functions ƒ and g that also satisfy these conditions. For instance,

ƒ(x)= (x – 5)³ – 4(x – 5) + 3     and    g(x) = x².

Related Answered Questions

Question: 2.5.8

Solving an Equation with a Graphing Calculator Use a graphing calculator to solve -2x – 4(2 – x) = 3x + 4. ...

Verified Answer:

We write an equivalent equation with 0 on one side...
Question: 2.7.3

Testing for Symmetry with Respect to an Axis Test for symmetry with respect to the x-axis and the y-axis. (a) y = x² + 4 (b) x = y² – 3 (c) x² + y² = 16 (d) 2x + y = 4 ...

Verified Answer:

(a) In y = x² + 4, replace x with -x. \lef...
Question: 2.8.8

Showing That (g ° f )(x) Is Not Equivalent to (f ° g) (x) Let ƒ(x) = 4x + 1 and g(x) = 2x² + 5x. Show that (g ∘ ƒ)(x) ≠ (ƒ ∘ g)(x). (This is sufficient to prove that this inequality is true in general.)  ...

Verified Answer:

First, find (g ∘ ƒ)(x). Then find (ƒ ∘ g)(x). [lat...
Question: 2.8.4

Finding the Difference Quotient Let ƒ(x) = 2x² – 3x. Find and simplify the expression for the difference quotient, ƒ(x + h) – ƒ(x)/h . ...

Verified Answer:

We use a three-step process. Step 1 Find the first...
Question: 2.8.3

Evaluating Combinations of Functions If possible, use the given representations of functions ƒ and g to evaluate (ƒ + g)(4), (ƒ – g)(-2), (ƒg)(1), and ( f/g )(0). (a) ...

Verified Answer:

(a) From the figure, ƒ(4) = 9 and g(4) = 2. (ƒ + g...
Question: 2.8.2

Using Operations on Functions and Determining Domains Let ƒ(x) = 8x – 9 and g(x) = √2x – 1. Find each function in (a)–(d). (a) (ƒ + g)(x) (b) (ƒ – g)(x) (c) (ƒg)(x) (d) ( f/g)(x) (e) Give the domains of the functions in parts (a)–(d). ...

Verified Answer:

(a) (ƒ + g)(x)                                    ...
Question: 2.8.1

Using Operations on Functions Let ƒ(x) = x² + 1 and g(x) = 3x + 5. Find each of the following. (a) (ƒ + g)(1) (b) (ƒ – g)(-3) (c) (ƒg)(5) (d) ( ƒ/g)(0) ...

Verified Answer:

(a) First determine ƒ(1) = 2 and g(1) = 8. Then us...
Question: 2.7.4

Testing for Symmetry with Respect to the Origin Determine whether the graph of each equation is symmetric with respect to the origin. (a) x² + y² = 16 (b) y = x³ ...

Verified Answer:

(a) Replace x with -x and y with -y. \left....
Question: 2.8.7

Determining Composite Functions and Their Domains Given that ƒ(x) = 6/x – 3 and g(x) = 1/x , find each of the following. (a) (ƒ ∘ g)(x) and its domain (b) (g ∘ ƒ)(x)and its domain ...

Verified Answer:

(a) (ƒ ∘ g)(x) = ƒ(g(x))        By definition [lat...
Question: 2.8.6

Determining Composite Functions and Their Domains Given that ƒ(x) = √x and g(x) = 4x + 2, find each of the following. (a) (ƒ ∘ g)(x) and its domain (b) (g ∘ ƒ)(x) and its domain ...

Verified Answer:

(a) (ƒ ∘ g)(x) = ƒ(g(x))          Definition of co...