Answer :
Given [tex]\( f(x) = x^2 + 1 \)[/tex] and [tex]\( g(x) = x - 4 \)[/tex], we need to find possible values of [tex]\( x \)[/tex] for which [tex]\( f(g(x)) \)[/tex] matches the given choices: 37, 97, 126, and 606.
We start by calculating [tex]\( f(g(x)) \)[/tex].
First, substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ g(x) = x - 4 \][/tex]
[tex]\[ f(g(x)) = f(x - 4) \][/tex]
Now calculate [tex]\( f(x - 4) \)[/tex]:
[tex]\[ f(x - 4) = (x - 4)^2 + 1 \][/tex]
Expand the expression:
[tex]\[ (x - 4)^2 + 1 = (x^2 - 8x + 16) + 1 = x^2 - 8x + 17 \][/tex]
So, we need to solve [tex]\( x^2 - 8x + 17 = \text{value} \)[/tex] for the given values.
1. First, solving for 37:
[tex]\[ x^2 - 8x + 17 = 37 \][/tex]
[tex]\[ x^2 - 8x + 17 - 37 = 0 \][/tex]
[tex]\[ x^2 - 8x - 20 = 0 \][/tex]
Solve the quadratic equation:
[tex]\[ x = \frac{8 \pm \sqrt{64 + 80}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{144}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm 12}{2} \][/tex]
[tex]\[ x = \frac{20}{2} = 10 \quad \text{or} \quad x = \frac{-4}{2} = -2 \][/tex]
Thus, [tex]\( x = 10 \)[/tex] or [tex]\( x = -2 \)[/tex] will satisfy [tex]\( f(g(x)) = 37 \)[/tex].
2. Second, solving for 97:
[tex]\[ x^2 - 8x + 17 = 97 \][/tex]
[tex]\[ x^2 - 8x + 17 - 97 = 0 \][/tex]
[tex]\[ x^2 - 8x - 80 = 0 \][/tex]
Solve the quadratic equation:
[tex]\[ x = \frac{8 \pm \sqrt{64 + 320}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{384}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm 8\sqrt{6}}{2} \][/tex]
[tex]\[ x = 4 \pm 4\sqrt{6} \][/tex]
Since [tex]\(\sqrt{6}\)[/tex] is not an integer, there are no integer solutions for [tex]\( x \)[/tex] here.
3. Third, solving for 126:
[tex]\[ x^2 - 8x + 17 = 126 \][/tex]
[tex]\[ x^2 - 8x + 17 - 126 = 0 \][/tex]
[tex]\[ x^2 - 8x - 109 = 0 \][/tex]
Solve the quadratic equation:
[tex]\[ x = \frac{8 \pm \sqrt{64 + 436}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{500}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm 10\sqrt{5}}{2} \][/tex]
[tex]\[ x = 4 \pm 5\sqrt{5} \][/tex]
Since [tex]\(\sqrt{5}\)[/tex] is not an integer, there are no integer solutions for [tex]\( x \)[/tex] here.
4. Finally, solving for 606:
[tex]\[ x^2 - 8x + 17 = 606 \][/tex]
[tex]\[ x^2 - 8x + 17 - 606 = 0 \][/tex]
[tex]\[ x^2 - 8x - 589 = 0 \][/tex]
Solve the quadratic equation:
[tex]\[ x = \frac{8 \pm \sqrt{64 + 2356}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{2420}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm 2\sqrt{605}}{2} \][/tex]
[tex]\[ x = 4 \pm \sqrt{605} \][/tex]
Since [tex]\(\sqrt{605}\)[/tex] is not an integer, there are no integer solutions for [tex]\( x \)[/tex] here.
Thus, the only value that satisfies the equation for integer [tex]\( x \)[/tex] is 37, and it is equivalent to the values of [tex]\( x = 10 \)[/tex] and [tex]\( x = -2 \)[/tex].
We start by calculating [tex]\( f(g(x)) \)[/tex].
First, substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ g(x) = x - 4 \][/tex]
[tex]\[ f(g(x)) = f(x - 4) \][/tex]
Now calculate [tex]\( f(x - 4) \)[/tex]:
[tex]\[ f(x - 4) = (x - 4)^2 + 1 \][/tex]
Expand the expression:
[tex]\[ (x - 4)^2 + 1 = (x^2 - 8x + 16) + 1 = x^2 - 8x + 17 \][/tex]
So, we need to solve [tex]\( x^2 - 8x + 17 = \text{value} \)[/tex] for the given values.
1. First, solving for 37:
[tex]\[ x^2 - 8x + 17 = 37 \][/tex]
[tex]\[ x^2 - 8x + 17 - 37 = 0 \][/tex]
[tex]\[ x^2 - 8x - 20 = 0 \][/tex]
Solve the quadratic equation:
[tex]\[ x = \frac{8 \pm \sqrt{64 + 80}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{144}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm 12}{2} \][/tex]
[tex]\[ x = \frac{20}{2} = 10 \quad \text{or} \quad x = \frac{-4}{2} = -2 \][/tex]
Thus, [tex]\( x = 10 \)[/tex] or [tex]\( x = -2 \)[/tex] will satisfy [tex]\( f(g(x)) = 37 \)[/tex].
2. Second, solving for 97:
[tex]\[ x^2 - 8x + 17 = 97 \][/tex]
[tex]\[ x^2 - 8x + 17 - 97 = 0 \][/tex]
[tex]\[ x^2 - 8x - 80 = 0 \][/tex]
Solve the quadratic equation:
[tex]\[ x = \frac{8 \pm \sqrt{64 + 320}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{384}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm 8\sqrt{6}}{2} \][/tex]
[tex]\[ x = 4 \pm 4\sqrt{6} \][/tex]
Since [tex]\(\sqrt{6}\)[/tex] is not an integer, there are no integer solutions for [tex]\( x \)[/tex] here.
3. Third, solving for 126:
[tex]\[ x^2 - 8x + 17 = 126 \][/tex]
[tex]\[ x^2 - 8x + 17 - 126 = 0 \][/tex]
[tex]\[ x^2 - 8x - 109 = 0 \][/tex]
Solve the quadratic equation:
[tex]\[ x = \frac{8 \pm \sqrt{64 + 436}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{500}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm 10\sqrt{5}}{2} \][/tex]
[tex]\[ x = 4 \pm 5\sqrt{5} \][/tex]
Since [tex]\(\sqrt{5}\)[/tex] is not an integer, there are no integer solutions for [tex]\( x \)[/tex] here.
4. Finally, solving for 606:
[tex]\[ x^2 - 8x + 17 = 606 \][/tex]
[tex]\[ x^2 - 8x + 17 - 606 = 0 \][/tex]
[tex]\[ x^2 - 8x - 589 = 0 \][/tex]
Solve the quadratic equation:
[tex]\[ x = \frac{8 \pm \sqrt{64 + 2356}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{2420}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm 2\sqrt{605}}{2} \][/tex]
[tex]\[ x = 4 \pm \sqrt{605} \][/tex]
Since [tex]\(\sqrt{605}\)[/tex] is not an integer, there are no integer solutions for [tex]\( x \)[/tex] here.
Thus, the only value that satisfies the equation for integer [tex]\( x \)[/tex] is 37, and it is equivalent to the values of [tex]\( x = 10 \)[/tex] and [tex]\( x = -2 \)[/tex].
