Answer :
Final answer:
For values of x greater than 5, y = x-5 is positive. For values of x less than 5, y = x-5 is negative. The function f(x) = |x-5|+3 can be rewritten as a piecewise function: f(x) = { x-5, if x < 5; |x-5|+3, if x ≥ 5 }. The graph of f(x) = |x-5|+3 is shown below:
Explanation:
To determine the values of x for which y = x-5 is positive, we need to consider the cases when (x-5) is positive or zero. When (x-5) is positive or zero, the absolute value function returns the input itself. Therefore, for y = x-5 to be positive, we need x-5 to be greater than zero.
Solving the inequality x-5 > 0, we add 5 to both sides to isolate x: x > 5.
So, for y = x-5 to be positive, x must be greater than 5.
To determine the values of x for which y = x-5 is negative, we need to consider the case when (x-5) is negative. When (x-5) is negative, the absolute value function returns the negation of the input. Therefore, for y = x-5 to be negative, we need x-5 to be less than zero.
Solving the inequality x-5 < 0, we add 5 to both sides to isolate x: x < 5.
So, for y = x-5 to be negative, x must be less than 5.
Now, let's rewrite f(x) as a piecewise function:
f(x) = { x-5, if x < 5; |x-5|+3, if x ≥ 5 }
Finally, let's graph the function f(x) = |x-5|+3:
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