Question 1

Let $$f(x) = x^2$ - 3x$$ and $$g(x) = 4x - 1. $$What is $$(f \circ g)(x)$$?

Accepted Answer

$$(4x - 1)^2$ - 3(4x - 1)$$

Question 2

Let $$f(x) = x^2$ - 3x$$ and $$g(x) = 4x - 1. $$What is $$(g \circ f)(x)$$?

Accepted Answer

$$4(x^2 - 3x) - 1$

Question 3

Let $$f(x) = \dfrac{2}{4 - x}$$ and $$g(x) = x^2$ + 1. $$What is $$(f \circ g)(x)$$?

Accepted Answer

$$\dfrac{2}{4 - (x^2$ + 1)}$$

Question 4

Let $$f(x) = \dfrac{2}{4 - x}$$ and $$g(x) = x^2$ + 1. $$What is $$(f \circ g)(3)$$?

Accepted Answer

$$\dfrac{2}{4 - (3^2$ + 1)} = \dfrac{2}{4 - 10} = \dfrac{2}{-6}$$

Question 5

Let $$f(x) = \dfrac{2}{4 - x}. $$What is $$(f \circ f)(2)$$?

Accepted Answer

$$\dfrac{2}{4 - \left(\dfrac{2}{4 - 2}\right)}$$

Question 6

Let $$f(x) = 3x - 5$$ and $$g(x) = \sqrt{2x + 1}. $$What is $$(f \circ g)(x)$$?

Accepted Answer

$$3\sqrt{2x + 1} - 5$$

Question 7

For $$f(x) = 3x - 5$$ and $$g(x) = \sqrt{2x + 1}$$, what is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x \geq -\dfrac{1}{2}$$

Question 8

Let $$f(x) = \dfrac{2}{4 - x}$$ and $$g(x) = x^2$ + 1. $$What is the domain of $$(g \circ f)(x)$$?

Accepted Answer

$$x 
eq 4$$

Question 9

Which of the following correctly describes the composition $$(f \circ g)(x)$$ for any functions $$f$$ and $$g$$?

Accepted Answer

$$f(g(x))$$

Question 10

Suppose $$h(x) = \dfrac{4}{\sqrt{2x - 5}}. $$Which pair of functions $$f(x)$$ and $$g(x)$$ could be used so that $$h(x) = (f \circ g)(x)$$?

Accepted Answer

$$f(x) = \dfrac{4}{x},\ g(x) = \sqrt{2x - 5}$$

Question 11

Suppose $$h(x) = 2 + (3x + 1)^4$. Which pair of functions f(x$$)$$ and g(x$$)$$ could be used so that h(x) = (f$$ \circ g)(x)$$?

Accepted Answer

$$f(x) = 2 + x^4$,\ g(x) = 3x + 1$$

Question 12

A medical dosage is calculated by $$f(x) = 2x + 5 mg$$, where $$x is $$the patient's weight in kg. If the patient's weight is estimated by $$g(y) = 0.5y + 10$$, where $$y is $$age in years, what is the formula for the dosage as a function of age?

Accepted Answer

$$f(g(y)) = 2(0.5y + 10) + 5$$

Question 13

If $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$, what is (g$$ \circ f)(x)$$?

Accepted Answer

$$\sqrt{x^2$}$$

Question 14

If $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$, what is the domain of (g$$ \circ f)(x)$$?

Accepted Answer

All real numbers $$x$$

Question 15

A chemical reaction rate is modeled by $$f(x) = \dfrac{1}{x}$$, where $$x is $$the concentration in mol/L. If the concentration is determined by $$g(t) = 2t + 1$$, where $$t is $$time in hours, what is the reaction rate as a function of time?

Accepted Answer

$$f(g(t)) = \dfrac{1}{2t + 1}$$

Question 16

If $$f(x) = x^2$ - 3x$$ and $$g(x) = 4x - 1$$, what is $$(f \circ g)(2)$$?

Accepted Answer

$$(4 \cdot 2 - 1)^2$ - 3(4 \cdot 2 - 1) = (8 - 1)^2$ - 3(8 - 1) = 7^2$ - 3 \cdot 7 = 49 - 21 = 28$$

Question 17

If $$f(x) = x^2$ - 3x$$ and $$g(x) = 4x - 1$$, what is $$(g \circ f)(2)$$?

Accepted Answer

$$4(2^2 - 3$$ \cdot 2) - 1 = 4(4 - 6) - 1 = 4(-2) - 1 = -8 - 1 = -9$$

Step-by-Step Guidance for Composition and Decomposition of Functions

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