Question 1
Let f(x) = 2_x_2 – 4_x_ + 1 and g(x) = (_x_2 + 16)(1/2). If k is a negative number such that f(k) = 31, then what is the value of (f(g(k))?
- 5
- -81
- 31
- 25
- -35
Explanation: In order to find the value of f(g(k)), we will first need to find k. We are told that f(k) = 31, so we can write an expression for f(k) and solve for k.
f(x) = 2_x_2 – 4_x_ + 1
f(k) = 2_k_2 – 4_k_ + 1 = 31
Subtract 31 from both sides.
2_k_2 – 4_k –_ 30 = 0
Divide both sides by 2.
k_2 – 2_k – 15 = 0
Now, we can factor this by thinking of two numbers that multiply to give –15 and add to give –2. These two numbers are –5 and 3.
k_2 –2_k – 15 = (k – 5)(k + 3) = 0
We can set each factor equal to 0 to find the values for k.
k – 5 = 0
Add 5 to both sides.
k = 5
Now we set k + 3 = 0.
Subtract 3 from both sides.
k = –3
This means that k could be either 5 or –3. However, we are told that k is a negative number, which means k = –3.
Finally, we can evaluate the expression f(g(–3)). First we need to find g(–3).
g(x) = (_x_2 + 16)(1/2)
g(–3) = ((–3)2 + 16)(1/2)
= (9 + 16)(1/2)
= 25(1/2)
Raising something to the one-half power is the same as taking the square root.
25(1/2) = 5
Now that we know g(–3) = 5, we must find f(5).
f(5) = 2(5)2 – 4(5) + 1
= 2(25) – 20 + 1 = 31
The answer is 31.