Question

math 2: transformations, radicals, and rational functions assessment

1. which transformation represents function ( f(x) = sqrt{x + 2} - 3 )?

a. shift left 2, then down 3
b. shift right 2, then down 3
c. shift left 2, then up 3
d. shift right 2, then up 3

1. what is the horizontal asymptote of ( f(x) = \frac{1}{2x} )?

a. ( y = 1 )
b. ( y = -1 )
c. ( y = 0 )
d. no horizontal asymptote exists

1. solve the equation ( sqrt{2x + 1} = 5 )

a. ( x = 12 )
b. ( x = 6 )
c. ( x = -6 )
d. ( x = 24 )

1. what is the vertical asymptote of ( f(x) = \frac{3}{x + 2} )?

a. ( x = 3 )
b. ( x = 0 )
c. ( x = -2 )
d. ( x = 2 )

1. which function represents a reflection of ( f(x) = sqrt{x} ) over the x - axis?

a. ( f(x) = -sqrt{x} )
b. ( f(x) = sqrt{-x} )
c. ( f(x) = sqrt{x + 1} )
d. ( f(x) = sqrt{x - 1} )

1. identify the domain of ( f(x) = sqrt{x - 4} )

a.
b. ( x > 4 )
c. ( x geq 4 )
d. ( x leq 4 )

1. find the vertical asymptote(s) of ( f(x)=\frac{3}{(x + 3)(x - 8)} )

Explanation:

Let's solve question 3: Solve the equation $\sqrt{2x + 7}=5$

Step 1: Square both sides

To eliminate the square root, we square both sides of the equation.
$$(\sqrt{2x + 7})^2 = 5^2$$
This simplifies to:
$$2x + 7 = 25$$

Step 2: Subtract 7 from both sides

Subtract 7 from each side to isolate the term with $x$.
$$2x + 7 - 7 = 25 - 7$$
Simplifying gives:
$$2x = 18$$

Step 3: Divide by 2

Divide both sides by 2 to solve for $x$.
$$\frac{2x}{2}=\frac{18}{2}$$
Which simplifies to:
$$x = 9$$Wait, but looking at the options, option a is $x = 12$? Wait, maybe I misread the equation. Wait, the original equation in the problem: "Solve the equation $\sqrt{2x + 7}=5$" Wait, no, maybe the equation is $\sqrt{2x + 1}=5$? Wait, the user's image: "Solve the equation $\sqrt{2x + 7}=5$" no, looking at the options: a. $x = 12$, b. $x = 6$, c. $x=- 6$, d. $x = 24$. Wait, maybe the equation is $\sqrt{2x + 1}=5$? No, let's re - check.

Wait, if the equation is $\sqrt{2x+7}=5$, then:

1. Square both sides: $2x + 7=25$
2. Subtract 7: $2x=25 - 7=18$
3. Divide by 2: $x = 9$. But 9 is not in the options. So maybe the equation is $\sqrt{2x + 1}=5$? No, $2x+1 = 25\Rightarrow2x = 24\Rightarrow x = 12$. Ah! Maybe the equation is $\sqrt{2x + 1}=5$. Let's assume that (maybe a typo in the original problem, 1 instead of 7).

So let's redo with $\sqrt{2x + 1}=5$:

Step 1: Square both sides

$$(\sqrt{2x + 1})^2=5^2$$
$$2x + 1 = 25$$

Step 2: Subtract 1

$$2x=25 - 1$$
$$2x = 24$$

Step 3: Divide by 2

$$x=\frac{24}{2}=12$$

Answer:

a. $x = 12$