At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
Certainly! Let's solve the problem step by step.
We are given two functions [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex]:
[tex]\[ g(x) = x + 1 \][/tex]
[tex]\[ h(x) = x^2 \][/tex]
We need to find the values of [tex]\( x \)[/tex] for which [tex]\( h(g(x)) = 3x^2 + x - 5 \)[/tex].
1. Find [tex]\( h(g(x)) \)[/tex]:
Start by substituting [tex]\( g(x) \)[/tex] into [tex]\( h(x) \)[/tex]:
[tex]\[ g(x) = x + 1 \][/tex]
[tex]\[ h(g(x)) = h(x + 1) \][/tex]
Since [tex]\( h(x) = x^2 \)[/tex], substitute [tex]\( x + 1 \)[/tex] into [tex]\( x \)[/tex] in [tex]\( h(x) \)[/tex]:
[tex]\[ h(x + 1) = (x + 1)^2 \][/tex]
2. Expand the expression:
Expand [tex]\( (x + 1)^2 \)[/tex]:
[tex]\[ (x + 1)^2 = x^2 + 2x + 1 \][/tex]
So, [tex]\( h(g(x)) = x^2 + 2x + 1 \)[/tex].
3. Set up the equation:
We are given that [tex]\( h(g(x)) = 3x^2 + x - 5 \)[/tex]. Substitute [tex]\( h(g(x)) \)[/tex] with [tex]\( x^2 + 2x + 1 \)[/tex]:
[tex]\[ x^2 + 2x + 1 = 3x^2 + x - 5 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
Move all terms to one side of the equation to set it to zero:
[tex]\[ x^2 + 2x + 1 - 3x^2 - x + 5 = 0 \][/tex]
Simplify the equation:
[tex]\[ -2x^2 + x + 6 = 0 \][/tex]
To make it more standard, multiply through by -1:
[tex]\[ 2x^2 - x - 6 = 0 \][/tex]
5. Factorize the quadratic equation:
We need to factor the equation [tex]\( 2x^2 - x - 6 = 0 \)[/tex]. To do this, we can use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 2 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = -6 \)[/tex].
First, calculate the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-1)^2 - 4(2)(-6) \][/tex]
[tex]\[ \Delta = 1 + 48 \][/tex]
[tex]\[ \Delta = 49 \][/tex]
The discriminant is positive, so we have two real solutions. Now substitute back into the quadratic formula:
[tex]\[ x = \frac{-(-1) \pm \sqrt{49}}{2(2)} \][/tex]
[tex]\[ x = \frac{1 \pm 7}{4} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{1 + 7}{4} = \frac{8}{4} = 2 \][/tex]
[tex]\[ x = \frac{1 - 7}{4} = \frac{-6}{4} = -\frac{3}{2} \][/tex]
Therefore, the values of [tex]\( x \)[/tex] for which [tex]\( h(g(x)) = 3x^2 + x - 5 \)[/tex] are:
[tex]\[ x = -\frac{3}{2} \quad \text{and} \quad x = 2 \][/tex]
We are given two functions [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex]:
[tex]\[ g(x) = x + 1 \][/tex]
[tex]\[ h(x) = x^2 \][/tex]
We need to find the values of [tex]\( x \)[/tex] for which [tex]\( h(g(x)) = 3x^2 + x - 5 \)[/tex].
1. Find [tex]\( h(g(x)) \)[/tex]:
Start by substituting [tex]\( g(x) \)[/tex] into [tex]\( h(x) \)[/tex]:
[tex]\[ g(x) = x + 1 \][/tex]
[tex]\[ h(g(x)) = h(x + 1) \][/tex]
Since [tex]\( h(x) = x^2 \)[/tex], substitute [tex]\( x + 1 \)[/tex] into [tex]\( x \)[/tex] in [tex]\( h(x) \)[/tex]:
[tex]\[ h(x + 1) = (x + 1)^2 \][/tex]
2. Expand the expression:
Expand [tex]\( (x + 1)^2 \)[/tex]:
[tex]\[ (x + 1)^2 = x^2 + 2x + 1 \][/tex]
So, [tex]\( h(g(x)) = x^2 + 2x + 1 \)[/tex].
3. Set up the equation:
We are given that [tex]\( h(g(x)) = 3x^2 + x - 5 \)[/tex]. Substitute [tex]\( h(g(x)) \)[/tex] with [tex]\( x^2 + 2x + 1 \)[/tex]:
[tex]\[ x^2 + 2x + 1 = 3x^2 + x - 5 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
Move all terms to one side of the equation to set it to zero:
[tex]\[ x^2 + 2x + 1 - 3x^2 - x + 5 = 0 \][/tex]
Simplify the equation:
[tex]\[ -2x^2 + x + 6 = 0 \][/tex]
To make it more standard, multiply through by -1:
[tex]\[ 2x^2 - x - 6 = 0 \][/tex]
5. Factorize the quadratic equation:
We need to factor the equation [tex]\( 2x^2 - x - 6 = 0 \)[/tex]. To do this, we can use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 2 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = -6 \)[/tex].
First, calculate the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-1)^2 - 4(2)(-6) \][/tex]
[tex]\[ \Delta = 1 + 48 \][/tex]
[tex]\[ \Delta = 49 \][/tex]
The discriminant is positive, so we have two real solutions. Now substitute back into the quadratic formula:
[tex]\[ x = \frac{-(-1) \pm \sqrt{49}}{2(2)} \][/tex]
[tex]\[ x = \frac{1 \pm 7}{4} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{1 + 7}{4} = \frac{8}{4} = 2 \][/tex]
[tex]\[ x = \frac{1 - 7}{4} = \frac{-6}{4} = -\frac{3}{2} \][/tex]
Therefore, the values of [tex]\( x \)[/tex] for which [tex]\( h(g(x)) = 3x^2 + x - 5 \)[/tex] are:
[tex]\[ x = -\frac{3}{2} \quad \text{and} \quad x = 2 \][/tex]
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
