Two turtles, Velma and Justine, were entered into a race.

Question

The equation y = 4x represents Velma's distance in meters, y, after racing for x minutes. The graph below displays Justine's distance and time.

Justine get 2 meters in 1 minute

Based on the equation and graph, which two statements below are true?

A. Justine is twice as fast as Velma.
B. Justine is half as fast as Velma.
C. Justine and Velma have the same speed.
D. Justine moves a greater distance than Velma each minute.
E. Justine moves a shorter distance than Velma each minute.

Answer 1

Let's analyze the information given concerning the turtles, Velma and Justine.

1. Velma's Speed: The equation \( y = 4x \) indicates Velma's distance traveled in meters in relation to time in minutes. From this equation, we can determine that Velma's speed is 4 meters per minute.

2. Justine's Speed: The problem states that Justine gets 2 meters in 1 minute. Therefore, Justine's speed is 2 meters per minute.

Now, comparing their speeds:

• Velma: 4 meters per minute
• Justine: 2 meters per minute

From these calculations, we can evaluate the statements:

A. Justine is twice as fast as Velma.

• This statement is false. Justine is slower than Velma.

B. Justine is half as fast as Velma.

• This statement is true. Since Velma's speed is 4 m/min and Justine's is 2 m/min, Justine's speed is indeed half of Velma's speed.

C. Justine and Velma have the same speed.

• This statement is false. They have different speeds.

D. Justine moves a greater distance than Velma each minute.

• This statement is false. Velma moves 4 meters, while Justine moves only 2 meters per minute.

E. Justine moves a shorter distance than Velma each minute.

• This statement is true. Justine moves 2 meters per minute, which is less than Velma's 4 meters.

Thus, the two true statements about their speeds are:

• B. Justine is half as fast as Velma.
• E. Justine moves a shorter distance than Velma each minute.

Answer 2

Given that k(x)is a one-to-one function and k^−1(x)is the inverse of k(x), if k^−1(12)=115and k^−1(15)=12, explore the function numerically to determine k(12).

Answer 3

Given the information about the function \( k \) and its inverse \( k^{-1} \), we can use the properties of inverse functions to determine \( k(12) \).

From the definition of inverse functions, we know:

• \( k^{-1}(y) = x \) implies \( k(x) = y \).

Now, let's analyze the given information:

1. From \( k^{-1}(12) = 115 \):
   This means that when \( k \) takes the input \( 115 \), it outputs \( 12 \). So we have: \[ k(115) = 12 \]

2. From \( k^{-1}(15) = 12 \):
   This means that when \( k \) takes the input \( 12 \), it outputs \( 15 \). So we have: \[ k(12) = 15 \]

Now, we can summarize what we’ve found:

• \( k(115) = 12 \)
• \( k(12) = 15 \)

Thus, the value of \( k(12) \) is \( \boxed{15} \).

Answer 4

Given that k(x)is a one-to-one function and k^−1(x)is the inverse of k(x), if k^−1(12)=1/15and k^−1(15)=12, explore the function numerically to determine k(12).

Answer 5

Given the properties of the function \( k(x) \) and its inverse \( k^{-1}(x) \), we can find \( k(12) \) using the provided information.

Information Given:

1. \( k^{-1}(12) = \frac{1}{15} \)
2. \( k^{-1}(15) = 12 \)

Understanding the Inverse Function:

Recall that for a function \( k \) and its inverse \( k^{-1} \):

• If \( k^{-1}(y) = x \), then \( k(x) = y \).

Applying the Information:

1. From \( k^{-1}(12) = \frac{1}{15} \): This means: \[ k\left(\frac{1}{15}\right) = 12 \]

2. From \( k^{-1}(15) = 12 \): This means: \[ k(12) = 15 \]

Conclusion:

Thus, the value of \( k(12) \) is: \[ \boxed{15} \]

Answer 6

what does \boxed{15}mean

Answer 7

The notation \(\boxed{15}\) is commonly used in mathematics as a way to highlight or emphasize a particular answer or result. The box around the number signifies that this is the key information or final answer to a problem.

In the context of our discussion, \(\boxed{15}\) indicates that the value of \( k(12) \) is \( 15 \). It is simply a way to make the answer stand out for clarity.

Answer 8

Given is the graph of the function h(x)=log_1/2 x. Which of the following correctly uses the graph to find h(1)+h^−1(−1)?

Answer 9

To solve \( h(1) + h^{-1}(-1) \) using the graph of the function \( h(x) = \log_{1/2}(x) \), we need to evaluate both components separately.

Step 1: Finding \( h(1) \)

The function \( h(x) = \log_{1/2}(x) \) represents the logarithm with a base of \( \frac{1}{2} \). The property of logarithms tells us that:

\[ h(1) = \log_{1/2}(1) \]

Since any number (including \(\frac{1}{2}\)) raised to the power of 0 equals 1:

\[ \log_{1/2}(1) = 0 \]

So, we find that:

\[ h(1) = 0 \]

Step 2: Finding \( h^{-1}(-1) \)

The value \( h^{-1}(-1) \) involves finding the \( x \) such that:

\[ h(x) = -1 \]

This means we need to solve:

\[ \log_{1/2}(x) = -1 \]

To convert the logarithmic equation to exponential form:

\[ x = \left( \frac{1}{2} \right)^{-1} \]

Calculating this gives:

\[ x = 2 \]

Therefore, we find that:

\[ h^{-1}(-1) = 2 \]

Step 3: Calculate \( h(1) + h^{-1}(-1) \)

Now, we can sum the values we found:

\[ h(1) + h^{-1}(-1) = 0 + 2 = 2 \]

Final Answer

Thus, the final result is:

\[ \boxed{2} \]