Final answer:
The absolute value function f(x) = |5x + 1| is expressed as a piecewise-defined function with f(x) = 5x + 1 for x ≥ -1/5 and f(x) = -(5x + 1) for x < -1/5.
Step-by-step explanation:
The absolute value function f(x) = |5x + 1| can be expressed as a piecewise-defined function with linear parts. To do this, we need to consider two cases: when the expression inside the absolute value is non-negative and when it's negative.
For the expression 5x + 1 to be non-negative, we need 5x + 1 ≥ 0, which gives x ≥ -1/5. For x values greater than or equal to -1/5, the absolute value function is just itself without the absolute value sign, since the expression inside is already positive or zero. Hence, for x ≥ -1/5:
f(x) = 5x + 1
For the expression 5x + 1 to be negative, we need 5x + 1 < 0, which gives x < -1/5. In this case, the value of the function is the negation of the expression inside the absolute value sign, to make it positive. Hence, for x < -1/5:
f(x) = -(5x + 1)
Putting it all together, we can write f(x) = |5x + 1| as:
- For x ≥ -1/5: f(x) = 5x + 1
- For x < -1/5: f(x) = -(5x + 1)