Question For the function $$ f(x)=|2 x-1|-3 $$, find the values of $$ x $$ such that $$ f(x)=0 $$.

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UpStudy AI · Accepted Answer

To find the values of $$ x $$ that satisfy the equation $$ f(x) = |2x - 1| - 3 = 0 $$, follow these steps:

1. **Set the Function Equal to Zero:**
   $$
   |2x - 1| - 3 = 0
   $$

2. **Isolate the Absolute Value:**
   $$
   |2x - 1| = 3
   $$
   
3. **Solve the Absolute Value Equation:**
   
   The equation $$ |A| = B $$ has two solutions:
   $$
   A = B \quad \text{or} \quad A = -B
   $$
   
   Apply this to $$ |2x - 1| = 3 $$:
   
   - **First Case:** 
     $$
     2x - 1 = 3
     $$
     $$
     2x = 4 \quad (\text{Add } 1 \text{ to both sides})
     $$
     $$
     x = 2 \quad (\text{Divide both sides by } 2)
     $$
   
   - **Second Case:** 
     $$
     2x - 1 = -3
     $$
     $$
     2x = -2 \quad (\text{Add } 1 \text{ to both sides})
     $$
     $$
     x = -1 \quad (\text{Divide both sides by } 2)
     $$
   
4. **Conclusion:**
   
   The solutions to the equation $$ f(x) = 0 $$ are:
   $$
   x = 2 \quad \text{and} \quad x = -1
   $$

**Verification:**

To ensure the solutions are correct, substitute them back into the original function.

- **For $$ x = 2 $$:**
  $$
  f(2) = |2(2) - 1| - 3 = |4 - 1| - 3 = |3| - 3 = 3 - 3 = 0
  $$
  
- **For $$ x = -1 $$:**
  $$
  f(-1) = |2(-1) - 1| - 3 = |-2 - 1| - 3 = |-3| - 3 = 3 - 3 = 0
  $$
  
Both values satisfy the equation $$ f(x) = 0 $$.

**Final Answer:**
$$
x = 2 \quad \text{and} \quad x = -1
$$
The values of $$ x $$ that satisfy $$ f(x) = 0 $$ are $$ x = 2 $$ and $$ x = -1 $$.

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For the function , find the values of such that .

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Question For the function , find the values of such that .