Join the IDNLearn.com community and start getting the answers you need today. Our community is here to provide the comprehensive and accurate answers you need to make informed decisions.
Sagot :
Sure, let's solve the problem step by step.
We are given two functions: [tex]\( f(x) = 3x + 4 \)[/tex] and [tex]\( g(x) = px + q \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are constants.
We need to determine the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] given the conditions: [tex]\( g(f(-1)) = 1 \)[/tex] and [tex]\( g^2(2) = 5 \)[/tex].
### Step 1: Calculate [tex]\( f(-1) \)[/tex]
First, we find the value of [tex]\( f(-1) \)[/tex]:
[tex]\[ f(x) = 3x + 4 \][/tex]
Substitute [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 3(-1) + 4 = -3 + 4 = 1 \][/tex]
### Step 2: Calculate [tex]\( g(f(-1)) \)[/tex]
We now calculate [tex]\( g(f(-1)) \)[/tex]:
[tex]\[ f(-1) = 1 \][/tex]
Therefore:
[tex]\[ g(f(-1)) = g(1) = p(1) + q = p + q \][/tex]
Given [tex]\( g(f(-1)) = 1 \)[/tex]:
[tex]\[ p + q = 1 \quad \text{(Equation 1)} \][/tex]
### Step 3: Calculate [tex]\( g^2(2) \)[/tex]
Next, we calculate [tex]\( g^2(2) \)[/tex] which means [tex]\( g(g(2)) \)[/tex]:
[tex]\[ g(x) = px + q \][/tex]
First, find [tex]\( g(2) \)[/tex]:
[tex]\[ g(2) = p(2) + q = 2p + q \][/tex]
Then calculate [tex]\( g(g(2)) \)[/tex]:
[tex]\[ g(g(2)) = g(2p + q) = p(2p + q) + q = 2p^2 + pq + q \][/tex]
Given [tex]\( g^2(2) = 5 \)[/tex]:
[tex]\[ 2p^2 + pq + q = 5 \quad \text{(Equation 2)} \][/tex]
### Step 4: Solve the system of equations
We have the following system of equations:
[tex]\[ \begin{cases} p + q = 1 & \text{(Equation 1)} \\ 2p^2 + pq + q = 5 & \text{(Equation 2)} \end{cases} \][/tex]
First, let's express [tex]\( q \)[/tex] from Equation 1:
[tex]\[ q = 1 - p \][/tex]
Now we substitute [tex]\( q \)[/tex] into Equation 2:
[tex]\[ 2p^2 + p(1 - p) + (1 - p) = 5 \][/tex]
Simplify it:
[tex]\[ 2p^2 + p - p^2 + 1 - p = 5 \][/tex]
[tex]\[ p^2 - p + 1 = 5 \][/tex]
[tex]\[ p^2 - p + 1 - 5 = 0 \][/tex]
[tex]\[ p^2 - p - 4 = 0 \][/tex]
Solve this quadratic equation:
[tex]\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Where [tex]\( a = 1 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = -4 \)[/tex]:
[tex]\[ p = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} \][/tex]
[tex]\[ p = \frac{1 \pm \sqrt{1 + 16}}{2} \][/tex]
[tex]\[ p = \frac{1 \pm \sqrt{17}}{2} \][/tex]
Thus, we have:
[tex]\[ p_1 = \frac{1 + \sqrt{17}}{2}, \quad p_2 = \frac{1 - \sqrt{17}}{2} \][/tex]
Now, find the corresponding [tex]\( q \)[/tex] values:
[tex]\[ q_1 = 1 - p_1 = 1 - \frac{1 + \sqrt{17}}{2} = \frac{2 - (1 + \sqrt{17})}{2} = \frac{1 - \sqrt{17}}{2} \][/tex]
[tex]\[ q_2 = 1 - p_2 = 1 - \frac{1 - \sqrt{17}}{2} = \frac{2 - (1 - \sqrt{17})}{2} = \frac{1 + \sqrt{17}}{2} \][/tex]
So, the possible pairs [tex]\((p, q)\)[/tex] are:
[tex]\[ \left( \frac{1 + \sqrt{17}}{2}, \frac{1 - \sqrt{17}}{2} \right) \quad \text{and} \quad \left( \frac{1 - \sqrt{17}}{2}, \frac{1 + \sqrt{17}}{2} \right) \][/tex]
We are given two functions: [tex]\( f(x) = 3x + 4 \)[/tex] and [tex]\( g(x) = px + q \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are constants.
We need to determine the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] given the conditions: [tex]\( g(f(-1)) = 1 \)[/tex] and [tex]\( g^2(2) = 5 \)[/tex].
### Step 1: Calculate [tex]\( f(-1) \)[/tex]
First, we find the value of [tex]\( f(-1) \)[/tex]:
[tex]\[ f(x) = 3x + 4 \][/tex]
Substitute [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 3(-1) + 4 = -3 + 4 = 1 \][/tex]
### Step 2: Calculate [tex]\( g(f(-1)) \)[/tex]
We now calculate [tex]\( g(f(-1)) \)[/tex]:
[tex]\[ f(-1) = 1 \][/tex]
Therefore:
[tex]\[ g(f(-1)) = g(1) = p(1) + q = p + q \][/tex]
Given [tex]\( g(f(-1)) = 1 \)[/tex]:
[tex]\[ p + q = 1 \quad \text{(Equation 1)} \][/tex]
### Step 3: Calculate [tex]\( g^2(2) \)[/tex]
Next, we calculate [tex]\( g^2(2) \)[/tex] which means [tex]\( g(g(2)) \)[/tex]:
[tex]\[ g(x) = px + q \][/tex]
First, find [tex]\( g(2) \)[/tex]:
[tex]\[ g(2) = p(2) + q = 2p + q \][/tex]
Then calculate [tex]\( g(g(2)) \)[/tex]:
[tex]\[ g(g(2)) = g(2p + q) = p(2p + q) + q = 2p^2 + pq + q \][/tex]
Given [tex]\( g^2(2) = 5 \)[/tex]:
[tex]\[ 2p^2 + pq + q = 5 \quad \text{(Equation 2)} \][/tex]
### Step 4: Solve the system of equations
We have the following system of equations:
[tex]\[ \begin{cases} p + q = 1 & \text{(Equation 1)} \\ 2p^2 + pq + q = 5 & \text{(Equation 2)} \end{cases} \][/tex]
First, let's express [tex]\( q \)[/tex] from Equation 1:
[tex]\[ q = 1 - p \][/tex]
Now we substitute [tex]\( q \)[/tex] into Equation 2:
[tex]\[ 2p^2 + p(1 - p) + (1 - p) = 5 \][/tex]
Simplify it:
[tex]\[ 2p^2 + p - p^2 + 1 - p = 5 \][/tex]
[tex]\[ p^2 - p + 1 = 5 \][/tex]
[tex]\[ p^2 - p + 1 - 5 = 0 \][/tex]
[tex]\[ p^2 - p - 4 = 0 \][/tex]
Solve this quadratic equation:
[tex]\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Where [tex]\( a = 1 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = -4 \)[/tex]:
[tex]\[ p = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} \][/tex]
[tex]\[ p = \frac{1 \pm \sqrt{1 + 16}}{2} \][/tex]
[tex]\[ p = \frac{1 \pm \sqrt{17}}{2} \][/tex]
Thus, we have:
[tex]\[ p_1 = \frac{1 + \sqrt{17}}{2}, \quad p_2 = \frac{1 - \sqrt{17}}{2} \][/tex]
Now, find the corresponding [tex]\( q \)[/tex] values:
[tex]\[ q_1 = 1 - p_1 = 1 - \frac{1 + \sqrt{17}}{2} = \frac{2 - (1 + \sqrt{17})}{2} = \frac{1 - \sqrt{17}}{2} \][/tex]
[tex]\[ q_2 = 1 - p_2 = 1 - \frac{1 - \sqrt{17}}{2} = \frac{2 - (1 - \sqrt{17})}{2} = \frac{1 + \sqrt{17}}{2} \][/tex]
So, the possible pairs [tex]\((p, q)\)[/tex] are:
[tex]\[ \left( \frac{1 + \sqrt{17}}{2}, \frac{1 - \sqrt{17}}{2} \right) \quad \text{and} \quad \left( \frac{1 - \sqrt{17}}{2}, \frac{1 + \sqrt{17}}{2} \right) \][/tex]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Your search for answers ends at IDNLearn.com. Thanks for visiting, and we look forward to helping you again soon.
