Question

F(x) =
4x + 5 if x ≤ 0
1
x + 2 if x > 0
4

Answer 1

The piecewise function
`\( f(x) \)` is defined as
`\( 4x + 5 \) for \( x \leq 0 \)` and
`\( (1)/(x + 2) \) for \( x > 0 \).` It exhibits a linear and rational behavior with a discontinuity at
`\( x = 0 \).`

The given function
`\( f(x) \)` is defined piecewise with distinct expressions for different intervals of
`\( x \).` For
`\( x \leq 0 \),` the function takes the form
`\( 4x + 5 \),`representing a linear equation with a slope of 4 and a y-intercept of 5. This portion of the function is characterized by a positive rate of change, implying that as
`\( x \)` decreases,
`\( f(x) \)` also decreases.

On the other hand, for
`\( x > 0 \),` the function adopts the expression
`\( (1)/(x + 2) \).` This is a rational function with a vertical asymptote at
`\( x = -2 \),` indicating a point of discontinuity in the function. As
`\( x \)` increases, the value of
`\( f(x) \)`decreases, approaching zero.

The piecewise structure accounts for the distinct behaviors of the function on either side of
`\( x = 0 \).`The function is continuous, but not differentiable at
`\( x = 0 \)`due to the sharp transition between the two expressions. Understanding the characteristics of each segment provides insight into the overall behavior of the function across its defined domain.