Answer :
Sure! To find \((f + g)(x)\) given the functions \(f(x) = -5^x - 4\) and \(g(x) = -3x - 2\), we follow these steps:
### Step 1: Define the Functions
First, let's rewrite the functions for clarity:
[tex]\[ f(x) = -5^x - 4 \][/tex]
[tex]\[ g(x) = -3x - 2 \][/tex]
### Step 2: Combine the Functions
To find \((f + g)(x)\), we need to sum \(f(x)\) and \(g(x)\). The combined function can be written as:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
[tex]\[ (f + g)(x) = (-5^x - 4) + (-3x - 2) \][/tex]
### Step 3: Simplify the Expression
Combine like terms to simplify the expression:
[tex]\[ (f + g)(x) = -5^x - 4 - 3x - 2 \][/tex]
[tex]\[ (f + g)(x) = -5^x - 3x - 6 \][/tex]
So, the function \((f + g)(x)\) is:
[tex]\[ (f + g)(x) = -5^x - 3x - 6 \][/tex]
### Step 4: Calculate Specific Values
Now let's calculate \((f + g)(x)\) for a range of \(x\) values, from \(x = -5\) to \(x = 5\).
- For \(x = -5\):
[tex]\[ (f + g)(-5) = -5^{-5} - 3(-5) - 6 = 0.00032 + 15 - 6 = 8.99968 \][/tex]
- For \(x = -4\):
[tex]\[ (f + g)(-4) = -5^{-4} - 3(-4) - 6 = 0.0016 + 12 - 6 = 5.9984 \][/tex]
- For \(x = -3\):
[tex]\[ (f + g)(-3) = -5^{-3} - 3(-3) - 6 = 0.008 - 9 - 6 = 2.992 \][/tex]
- For \(x = -2\):
[tex]\[ (f + g)(-2) = -5^{-2} - 3(-2) - 6 = 0.04 + 6 - 6 = -0.040000000000000036 \][/tex]
- For \(x = -1\):
[tex]\[ (f + g)(-1) = -5^{-1} - 3(-1) - 6 = 0.2 + 3 - 6 = -3.2 \][/tex]
- For \(x = 0\):
[tex]\[ (f + g)(0) = -5^0 - 3(0) - 6 = -1 - 0 - 6 = -7 \][/tex]
- For \(x = 1\):
[tex]\[ (f + g)(1) = -5^1 - 3(1) - 6 = -5 - 3 - 6 = -14 \][/tex]
- For \(x = 2\):
[tex]\[ (f + g)(2) = -5^2 - 3(2) - 6 = -25 - 6 - 6 = -37 \][/tex]
- For \(x = 3\):
[tex]\[ (f + g)(3) = -5^3 - 3(3) - 6 = -125 - 9 - 6 = -140 \][/tex]
- For \(x = 4\):
[tex]\[ (f + g)(4) = -5^4 - 3(4) - 6 = -625 - 12 - 6 = -643 \][/tex]
- For \(x = 5\):
[tex]\[ (f + g)(5) = -5^5 - 3(5) - 6 = -3125 - 15 - 6 = -3146 \][/tex]
### Summary of Results:
Here are the results of \((f + g)(x)\) for the given \(x\) values:
- \((f + g)(-5) = 8.99968\)
- \((f + g)(-4) = 5.9984\)
- \((f + g)(-3) = 2.992\)
- \((f + g)(-2) = -0.040000000000000036\)
- \((f + g)(-1) = -3.2\)
- \((f + g)(0) = -7\)
- \((f + g)(1) = -14\)
- \((f + g)(2) = -37\)
- \((f + g)(3) = -140\)
- \((f + g)(4) = -643\)
- \((f + g)(5) = -3146\)
These values show the combinations of the two functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] across the specified range.
### Step 1: Define the Functions
First, let's rewrite the functions for clarity:
[tex]\[ f(x) = -5^x - 4 \][/tex]
[tex]\[ g(x) = -3x - 2 \][/tex]
### Step 2: Combine the Functions
To find \((f + g)(x)\), we need to sum \(f(x)\) and \(g(x)\). The combined function can be written as:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
[tex]\[ (f + g)(x) = (-5^x - 4) + (-3x - 2) \][/tex]
### Step 3: Simplify the Expression
Combine like terms to simplify the expression:
[tex]\[ (f + g)(x) = -5^x - 4 - 3x - 2 \][/tex]
[tex]\[ (f + g)(x) = -5^x - 3x - 6 \][/tex]
So, the function \((f + g)(x)\) is:
[tex]\[ (f + g)(x) = -5^x - 3x - 6 \][/tex]
### Step 4: Calculate Specific Values
Now let's calculate \((f + g)(x)\) for a range of \(x\) values, from \(x = -5\) to \(x = 5\).
- For \(x = -5\):
[tex]\[ (f + g)(-5) = -5^{-5} - 3(-5) - 6 = 0.00032 + 15 - 6 = 8.99968 \][/tex]
- For \(x = -4\):
[tex]\[ (f + g)(-4) = -5^{-4} - 3(-4) - 6 = 0.0016 + 12 - 6 = 5.9984 \][/tex]
- For \(x = -3\):
[tex]\[ (f + g)(-3) = -5^{-3} - 3(-3) - 6 = 0.008 - 9 - 6 = 2.992 \][/tex]
- For \(x = -2\):
[tex]\[ (f + g)(-2) = -5^{-2} - 3(-2) - 6 = 0.04 + 6 - 6 = -0.040000000000000036 \][/tex]
- For \(x = -1\):
[tex]\[ (f + g)(-1) = -5^{-1} - 3(-1) - 6 = 0.2 + 3 - 6 = -3.2 \][/tex]
- For \(x = 0\):
[tex]\[ (f + g)(0) = -5^0 - 3(0) - 6 = -1 - 0 - 6 = -7 \][/tex]
- For \(x = 1\):
[tex]\[ (f + g)(1) = -5^1 - 3(1) - 6 = -5 - 3 - 6 = -14 \][/tex]
- For \(x = 2\):
[tex]\[ (f + g)(2) = -5^2 - 3(2) - 6 = -25 - 6 - 6 = -37 \][/tex]
- For \(x = 3\):
[tex]\[ (f + g)(3) = -5^3 - 3(3) - 6 = -125 - 9 - 6 = -140 \][/tex]
- For \(x = 4\):
[tex]\[ (f + g)(4) = -5^4 - 3(4) - 6 = -625 - 12 - 6 = -643 \][/tex]
- For \(x = 5\):
[tex]\[ (f + g)(5) = -5^5 - 3(5) - 6 = -3125 - 15 - 6 = -3146 \][/tex]
### Summary of Results:
Here are the results of \((f + g)(x)\) for the given \(x\) values:
- \((f + g)(-5) = 8.99968\)
- \((f + g)(-4) = 5.9984\)
- \((f + g)(-3) = 2.992\)
- \((f + g)(-2) = -0.040000000000000036\)
- \((f + g)(-1) = -3.2\)
- \((f + g)(0) = -7\)
- \((f + g)(1) = -14\)
- \((f + g)(2) = -37\)
- \((f + g)(3) = -140\)
- \((f + g)(4) = -643\)
- \((f + g)(5) = -3146\)
These values show the combinations of the two functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] across the specified range.
