Questions: Find points on the graph of the function (k(x)=-frac3 sqrt[4]x7), other than the origin, that fit within the given ([-10,10]) by ([-10,10]) to two decimal places as necessary.

Solution Steps

The given function is \( k(x) = -\frac{3 \sqrt[4]{x}}{7} \). This is a fourth root function scaled by a factor of \(-\frac{3}{7}\).

The domain of \( k(x) \) is \( x \geq 0 \) because the fourth root of a negative number is not a real number.

We need to find points within the range \([-10, 10]\) for both \( x \) and \( y \). Let's calculate \( k(x) \) for some values of \( x \) within this range.

• For \( x = 1 \): \[ k(1) = -\frac{3 \sqrt[4]{1}}{7} = -\frac{3 \cdot 1}{7} = -\frac{3}{7} \approx -0.43 \]
• For \( x = 16 \): \[ k(16) = -\frac{3 \sqrt[4]{16}}{7} = -\frac{3 \cdot 2}{7} = -\frac{6}{7} \approx -0.86 \]
• For \( x = 81 \): \[ k(81) = -\frac{3 \sqrt[4]{81}}{7} = -\frac{3 \cdot 3}{7} = -\frac{9}{7} \approx -1.29 \]

We need to ensure the points fit within the \([-10, 10]\) range for both \( x \) and \( y \).

• \( (1, -0.43) \)
• \( (16, -0.86) \)
• \( (81, -1.29) \)

Since \( x = 16 \) and \( x = 81 \) are outside the \([-10, 10]\) range, we discard them.

The valid point within the given range is:

• \( (1, -0.43) \)

Final Answer