Question
The function \( f \) is defined as follows. \[ f(x)=\left\{\begin{array}{l}|2 x| \text { if }-3 \leq x<0 \\ x^{3} \text { if } x \geq 0\end{array}\right. \] (a) Find the domain of the function. (b) Locate any intercepts. (c) Graph the function. (d) Based on the graph, find the range. a) The domain of the function \( f \) is Type your answer in interval notation.)
Ask by Sullivan Davies. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The domain of the function \( f \) is \([-3, \infty)\).
Solution
We are given the function
f(x) = |2x| if –3 ≤ x < 0
f(x) = x³ if x ≥ 0
For the domain, notice that:
• The first piece is defined for all x between –3 and 0 (including –3, but not 0).
• The second piece is defined for all x greater than or equal to 0.
Thus, f(x) is defined for every x starting at –3 and continuing to positive infinity. In interval notation, the domain is:
[–3, ∞)
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The domain of the function \( f \) is \([-3, \infty)\). This is because for the first piece, \( |2x| \), the function is defined for \( -3 \leq x < 0 \), and for the second piece, \( x^3 \), it is defined for \( x \geq 0\). Combining these, we include all values from -3 to infinity. Now, let's move on to intercepts. The \( y \)-intercept occurs when \( x = 0 \), so \( f(0) = 0^3 = 0 \). The \( x \)-intercept is when \( f(x) = 0 \). In the first piece, \( |2x| = 0 \) when \( x = 0 \). In the second piece, \( x^3 = 0 \) also gives \( x = 0 \). Therefore, the only intercept is at \( (0, 0) \).
