The graph of f(x) = |x| is reflected across the y-axis and translated to the left 5 units. Which statement about the domain and range of each function is correct?

Question

Both the domain and range of the transformed function are the same as those of the parent function.
Neither the domain nor the range of the transformed function are the same as those of the parent function.
The range of the transformed function is the same as the parent function, but the domains of the functions are different.
The domain of the transformed function is the same as the parent function, but the ranges of the functions are different.

Answer 1

To analyze the transformation of the function \( f(x) = |x| \), let's first establish the domain and range of the parent function.

For the parent function \( f(x) = |x| \):

• Domain: All real numbers, or \( (-\infty, \infty) \)
• Range: All non-negative real numbers, or \( [0, \infty) \)

Now, let's handle the transformations:

1. Reflection across the y-axis: When \( f(x) = |x| \) is reflected across the y-axis, it remains \( f(-x) = |-x| = |x| \). Thus, this transformation does not change the function.

2. Translation to the left by 5 units: This transformation changes the function to \( g(x) = |x + 5| \).

Now we can analyze the domain and range of \( g(x) \).

For the transformed function \( g(x) = |x + 5| \):

• Domain: All real numbers, or \( (-\infty, \infty) \) (the same as the parent function)
• Range: All non-negative real numbers, or \( [0, \infty) \) (the same as the parent function)

Since both the domain and range of \( g(x) \) are the same as those of \( f(x) \), the correct statement is:

Both the domain and range of the transformed function are the same as those of the parent function.

Answer 2

The absolute value function, f(x) = –|x| – 3, is shown.What is the range of the function?
all real numbers
all real numbers less than or equal to 0
all real numbers greater than or equal to –3
all real numbers less than or equal to –3

Answer 3

To find the range of the function \( f(x) = -|x| - 3 \), let's analyze it step by step.

1. Understanding the Absolute Value Function: The absolute value function, \( |x| \), outputs non-negative values (i.e., \( |x| \geq 0 \) for all \( x \)).

2. Applying the Negation: When you negate the absolute value, \( -|x| \), the result will be non-positive, meaning \( -|x| \leq 0 \). The maximum value of \( -|x| \) is 0, which occurs when \( x = 0 \) (since \( |x| = 0 \)).

3. Subtracting 3: Next, we subtract 3 from \( -|x| \): \[ f(x) = -|x| - 3 \] This will shift the entire output down by 3 units. Thus, the maximum value of \( f(x) \) becomes \( 0 - 3 = -3 \).

4. Determining the Range: Since \( -|x| \) can take any value \( \leq 0 \), and after subtracting 3 it becomes \( -3 \) or any lower number. Therefore, the function can take any value less than or equal to \(-3\).

Thus, the range of the function \( f(x) = -|x| - 3 \) is:

All real numbers less than or equal to -3.

Answer 4

The absolute value function, f(x) = |x + 2|, is shown.

If the graph is reflected over the x-axis, what is the domain of the function?

all real numbers
all real numbers greater than or equal to 0
all real numbers greater than or equal to –2
all real numbers less than or equal to –2

Answer 5

To determine the domain of the function after reflecting \( f(x) = |x + 2| \) over the x-axis, let's analyze both the original function and the transformed function.

1. Original Function: The function \( f(x) = |x + 2| \) has the following domain:

   • Domain: All real numbers, \( (-\infty, \infty) \).

2. Reflection Over the X-Axis: When we reflect the function over the x-axis, it changes to: \[ g(x) = -|x + 2| \] Reflecting over the x-axis does not affect the x-values; it only changes the output of the function.

3. Domain of the Transformed Function: Since the x-values (inputs) are unchanged by the reflection, the domain of \( g(x) = -|x + 2| \) remains the same as that of \( f(x) = |x + 2| \).

Thus, the domain of the function \( g(x) = -|x + 2| \) after reflecting over the x-axis is still:

All real numbers.

Answer 6

i got 60.

The function shown is reflected across the y-axis to create a new function.

Which is true about the domain and range of each function?

Both the domain and range change.
Both the range and domain stay the same.
The domain stays the same, but the range changes.
The range stays the same, but the domain changes.

Answer 7

Let’s analyze the situation: you have a function that is reflected across the y-axis to create a new function.

1. Reflection Across the Y-Axis: When a function \( f(x) \) is reflected across the y-axis, the new function is represented as \( f(-x) \). This reflection process influences the input values of the function (the x-values), but it does not alter the output values within the range.

2. Domain: The domain of the original function \( f(x) \) will generally change after reflection, depending on the specific function. However, for many functions that are defined for all real numbers (like linear functions, polynomial functions, etc.), the domain remains the same.

3. Range: The range of the function is determined by the output values of \( f(x) \) and is unaffected by how the function is represented lexically (whether as \( f(x) \) or \( f(-x) \)) since the output values are the same.

From these points, we can conclude that for many functions, reflecting across the y-axis generally preserves the range while potentially altering the domain based on the specific characteristics of the function.

Conclusion

Thus, the correct statement regarding the domain and range after the reflection is:

The range stays the same, but the domain changes.

Answer 8

The absolute value function, f(x) = –|x| – 3, is shown.

What is the range of the function?

all real numbers
all real numbers less than or equal to 0
all real numbers greater than or equal to –3
all real numbers less than or equal to –3

Answer 9

To determine the range of the function \( f(x) = -|x| - 3 \), let's analyze it step by step.

1. Understanding the Absolute Value Function: The graph of \( |x| \) is a V-shape centered at the origin, and it always outputs non-negative values, so \( |x| \geq 0 \) for all \( x \).

2. Applying the Negation: The term \( -|x| \) will then be less than or equal to zero, since: \[ -|x| \leq 0 \]

3. Subtracting 3: Next, we calculate what happens when we subtract 3 from \( -|x| \): \[ f(x) = -|x| - 3 \] The maximum value of \( -|x| \) is \( 0 \) (which occurs when \( x = 0 \)), and when we subtract 3, this gives us: \[ f(0) = -|0| - 3 = 0 - 3 = -3 \]

4. Determining the Range: As \( |x| \) increases (as \( x \) moves away from 0), \( -|x| \) decreases (becomes more negative). Therefore, \( -|x| - 3 \) will always yield values less than or equal to \(-3\).

So, the range of \( f(x) = -|x| - 3 \) is:

All real numbers less than or equal to -3.