Questions: Use the piecewise-defined function to find the following values for (f(x)). (f(x)=leftbeginarrayll 1-3 x text if x leq 1 3 x text if 1<x<7 4 x+3 text if x geq 7 endarrayright.) Find (f(-1), f(1), f(4), f(6)), and (f(7)).

Solution Steps

To find the values of the piecewise-defined function $f(x)$ at specific points, we need to determine which piece of the function applies to each given value of $x$. We then substitute the value of $x$ into the appropriate expression to calculate $f(x)$.

1. For $f(-1)$, since $-1 \leq 1$, use the first piece: $f(x) = 1 - 3x$.
2. For $f(1)$, since $1 \leq 1$, use the first piece: $f(x) = 1 - 3x$.
3. For $f(4)$, since $1 < 4 < 7$, use the second piece: $f(x) = 3x$.
4. For $f(6)$, since $1 < 6 < 7$, use the second piece: $f(x) = 3x$.
5. For $f(7)$, since $7 \geq 7$, use the third piece: $f(x) = 4x + 3$.

For $x = -1$, since $-1 \leq 1$, we use the first piece of the function: $$f(-1) = 1 - 3(-1) = 1 + 3 = 4$$

For $x = 1$, since $1 \leq 1$, we again use the first piece of the function: $$f(1) = 1 - 3(1) = 1 - 3 = -2$$

For $x = 4$, since $1 < 4 < 7$, we use the second piece of the function: $$f(4) = 3(4) = 12$$

For $x = 6$, since $1 < 6 < 7$, we again use the second piece of the function: $$f(6) = 3(6) = 18$$

For $x = 7$, since $7 \geq 7$, we use the third piece of the function: $$f(7) = 4(7) + 3 = 28 + 3 = 31$$

Final Answer