Answer :
To determine the possible domains for the function [tex]\( f(x) = |x + 7| - 5 \)[/tex] that will make it invertible, let's follow these steps:
1. Understand the Function: The function [tex]\( f(x) = |x + 7| - 5 \)[/tex] is an absolute value function shifted to the left by 7 units and down by 5 units.
2. Graph and Vertex: The absolute value function [tex]\( |x + 7| \)[/tex] has its vertex at [tex]\( x = -7 \)[/tex]. When we subtract 5, the function is simply shifted down vertically but the vertex at [tex]\( x = -7 \)[/tex] remains.
3. Invertibility Criteria: For a function to be invertible, it must be one-to-one, which means it should pass the Horizontal Line Test. An absolute value function is typically not one-to-one because it is symmetrical around its vertex and has both increasing and decreasing parts.
4. Restricting the Domain:
- To make the function one-to-one, we need to restrict its domain to either the left side or the right side of the vertex.
- The vertex of the function is [tex]\( x = -7 \)[/tex]. We can choose the domain to be either [tex]\( x \leq -7 \)[/tex] or [tex]\( x \geq -7 \)[/tex]. Within these ranges, the function will be either strictly increasing or strictly decreasing.
5. Conclusion:
- The possible domains that will make the function [tex]\( f(x) = |x + 7| - 5 \)[/tex] invertible are [tex]\( x \leq -7 \)[/tex] and [tex]\( x \geq -7 \)[/tex].
### Final Answer:
The possible domains for the function [tex]\( f(x) = |x + 7| - 5 \)[/tex] to be invertible are:
[tex]\[ x \leq -7 \quad \text{or} \quad x \geq -7 \][/tex]
1. Understand the Function: The function [tex]\( f(x) = |x + 7| - 5 \)[/tex] is an absolute value function shifted to the left by 7 units and down by 5 units.
2. Graph and Vertex: The absolute value function [tex]\( |x + 7| \)[/tex] has its vertex at [tex]\( x = -7 \)[/tex]. When we subtract 5, the function is simply shifted down vertically but the vertex at [tex]\( x = -7 \)[/tex] remains.
3. Invertibility Criteria: For a function to be invertible, it must be one-to-one, which means it should pass the Horizontal Line Test. An absolute value function is typically not one-to-one because it is symmetrical around its vertex and has both increasing and decreasing parts.
4. Restricting the Domain:
- To make the function one-to-one, we need to restrict its domain to either the left side or the right side of the vertex.
- The vertex of the function is [tex]\( x = -7 \)[/tex]. We can choose the domain to be either [tex]\( x \leq -7 \)[/tex] or [tex]\( x \geq -7 \)[/tex]. Within these ranges, the function will be either strictly increasing or strictly decreasing.
5. Conclusion:
- The possible domains that will make the function [tex]\( f(x) = |x + 7| - 5 \)[/tex] invertible are [tex]\( x \leq -7 \)[/tex] and [tex]\( x \geq -7 \)[/tex].
### Final Answer:
The possible domains for the function [tex]\( f(x) = |x + 7| - 5 \)[/tex] to be invertible are:
[tex]\[ x \leq -7 \quad \text{or} \quad x \geq -7 \][/tex]
