Question

The function f(x)\:=\:x^{\frac{1}{3}} is transformed to get function h. which statements are true about function h?

A) The transformation involves a vertical stretch.
B) The transformation involves a horizontal compression.
C) The transformation results in a reflection across the y-axis.
D) The transformation shifts the graph vertically.

Answer 1

The correct statements about function h are B) The transformation involves a horizontal compression and D) The transformation shifts the graph vertically.

Step-by-step explanation:

To analyze the transformation of
`\(f(x) = x^{(1)/(3)}\)` to function h, let's consider each statement.

B) Horizontal Compression:

A horizontal compression is achieved by multiplying the variable
`\(x\)` inside the function by a constant between 0 and 1. The general form of a horizontally compressed function is
`\(h(x) = c \cdot f(x)\)`, where
`\(c < 1\)`. In this case, the cube root function
`\(f(x)\)` is horizontally compressed if
`\(h(x) = k \cdot x^{(1)/(3)}\), where \(k < 1\).`

D) Vertical Shift:

A vertical shift is accomplished by adding or subtracting a constant outside the function. The general form of a vertically shifted function is
`\(h(x) = f(x) + d\)`, where
`\(d\)` is a constant. In the context of
`\(f(x) = x^{(1)/(3)}\)`, the vertical shift is represented by
`\(h(x) = x^{(1)/(3)} + a\)`, where \(a\) is the vertical shift.

Therefore, the correct statements are B) The transformation involves a horizontal compression, indicating the presence of a constant multiplier inside the function, and D) The transformation shifts the graph vertically, suggesting the addition or subtraction of a constant outside the function.