If `p (x) = x ^(2) -4x+8 and q 9x)=x-3,` what is the value of `(q (p (5)))/(p (q (5)))` ?

Options

A. 0
B. `0.4`
C. 1
D. `2.5`

Explanation

To solve the problem, we need to find the value of \(\frac{q(p(5))}{p(q(5))}\) given the functions \(p(x) = x^2 - 4x + 8\) and \(q(x) = x - 3\).

### Step 1: Calculate \(p(5)\)

Using the function \(p(x)\):
\[ p(5) = 5^2 - 4 \cdot 5 + 8 \]
Calculating this step-by-step:
- \(5^2 = 25\)
- \(4 \cdot 5 = 20\)
- Now substitute these values:
\[ p(5) = 25 - 20 + 8 = 25 - 20 + 8 = 13 \]

### Step 2: Calculate \(q(p(5))\)

Now we need to find \(q(p(5))\), which is \(q(13)\):
\[ q(13) = 13 - 3 = 10 \]

### Step 3: Calculate \(q(5)\)

Next, we will calculate \(q(5)\):
\[ q(5) = 5 - 3 = 2 \]

### Step 4: Calculate \(p(q(5))\)

Now we need to find \(p(q(5))\), which is \(p(2)\):
\[ p(2) = 2^2 - 4 \cdot 2 + 8 \]
Calculating this step-by-step:
- \(2^2 = 4\)
- \(4 \cdot 2 = 8\)
- Now substitute these values:
\[ p(2) = 4 - 8 + 8 = 4 \]

### Step 5: Calculate \(\frac{q(p(5))}{p(q(5))}\)

Now we can find the final value:
\[ \frac{q(p(5))}{p(q(5))} = \frac{10}{4} = 2.5 \]

### Final Answer
Thus, the value of \(\frac{q(p(5))}{p(q(5))}\) is \(2.5\).

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