A function h is defined as follows :
for `x gt 0, h(x)=x^(7)+2x^(5)-12x^(3)+15x-2`
for `x le0, h(x)=x^(6)-3x^(4)+2x^(2)-7x-5`
What is the value of `h(-1)`?

Answer

To find the value of \( h(-1) \), we first need to determine which part of the function definition applies to \( x = -1 \).

### Step 1: Identify the correct function definition
Since \( -1 \leq 0 \), we use the definition of \( h(x) \) for \( x \leq 0 \):
\[ h(x) = x^6 - 3x^4 + 2x^2 - 7x - 5 \]

### Step 2: Substitute \( x = -1 \) into the function
Now, we substitute \( x = -1 \) into the equation:
\[ h(-1) = (-1)^6 - 3(-1)^4 + 2(-1)^2 - 7(-1) - 5 \]

### Step 3: Calculate each term
- Calculate \( (-1)^6 \):
\[ (-1)^6 = 1 \]
- Calculate \( -3(-1)^4 \):
\[ -3(-1)^4 = -3 \cdot 1 = -3 \]
- Calculate \( 2(-1)^2 \):
\[ 2(-1)^2 = 2 \cdot 1 = 2 \]
- Calculate \( -7(-1) \):
\[ -7(-1) = 7 \]
- The last term is \( -5 \).

### Step 4: Combine all the terms
Now, we combine all the calculated values:
\[ h(-1) = 1 - 3 + 2 + 7 - 5 \]

### Step 5: Simplify the expression
Now, we simplify:
\[ h(-1) = (1 - 3) + 2 + 7 - 5 \]
\[ = -2 + 2 + 7 - 5 \]
\[ = 0 + 7 - 5 \]
\[ = 7 - 5 = 2 \]

### Conclusion
Thus, the value of \( h(-1) \) is:
\[ \boxed{2} \]