Analyze the effect on the graph from g(x)=2x−1−−−−−√+2

Question

𝑔
(
𝑥
)
=
2
𝑥
−
1
+
2
to f(x)=2x−1−−−−−√+10
𝑓
(
𝑥
)
=
2
𝑥
−
1
+
10
. Which option correctly describes the effect?(1 point)
Responses

The range changes from [2,∞)
[
2
,
∞
)
to [10,∞)
[
10
,
∞
)
.
The range changes from left bracket 2 comma infinity right parenthesis to left bracket 10 comma infinity right parenthesis .

The domain changes from [1,∞)
[
1
,
∞
)
to [−1,∞)
[
−
1
,
∞
)
.
The domain changes from left bracket 1 comma infinity right parenthesis to left bracket negative 1 comma infinity right parenthesis .

The domain changes from [2,∞)
[
2
,
∞
)
to [10,∞)
[
10
,
∞
)
.
The domain changes from left bracket 2 comma infinity right parenthesis to left bracket 10 comma infinity right parenthesis .

The range changes from [2,∞)
[
2
,
∞
)
to [8,∞)
[
8
,
∞
)
.
The range changes from left bracket 2 comma infinity right pare

Answer 1

To analyze the effects on the function \( g(x) = \sqrt{2x - 1} + 2 \) when it changes to \( f(x) = \sqrt{2x - 1} + 10 \), let's consider the following aspects: the domain and the range.

Domain

The domain of a function is determined by the values of \( x \) for which the expression under the square root is non-negative.

For \( g(x) \):

• We set \( 2x - 1 \geq 0 \), which leads to \( 2x \geq 1 \) or \( x \geq \frac{1}{2} \).
  • Thus, the domain of \( g(x) \) is \( [\frac{1}{2}, \infty) \).

For \( f(x) \):

• The domain remains the same: \( 2x - 1 \geq 0 \) leads to the same conclusion \( x \geq \frac{1}{2} \).
  • Thus, the domain of \( f(x) \) is also \( [\frac{1}{2}, \infty) \).

Range

The range of the function can be determined by analyzing the outputs of the functions based on their minimum values.

For \( g(x) \):

• The minimum value occurs when \( x = \frac{1}{2} \): \[ g\left(\frac{1}{2}\right) = \sqrt{2(\frac{1}{2}) - 1} + 2 = \sqrt{0} + 2 = 2 \]
• Therefore, the range of \( g(x) \) is \( [2, \infty) \).

For \( f(x) \):

• The minimum value occurs at the same \( x = \frac{1}{2} \): \[ f\left(\frac{1}{2}\right) = \sqrt{2(\frac{1}{2}) - 1} + 10 = \sqrt{0} + 10 = 10 \]
• Therefore, the range of \( f(x) \) is \( [10, \infty) \).

Summary of Changes

• Domain: The domain remains the same \( [\frac{1}{2}, \infty) \).
• Range: The range changes from \( [2, \infty) \) for \( g(x) \) to \( [10, \infty) \) for \( f(x) \).

Correct Response

Based on this analysis, the correct option describing the effect is:

The range changes from \( [2, \infty) \) to \( [10, \infty) \).