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Solve the following problems involving algebraic equations and functions: a) Find g(1+x) given g(x) = x2 + 1. b) Find the range of x for which F(x) < -3 given F(x) = 2x − 1. c) Simplify F(x) * g(x). d

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4 parts
a

g(1+x)=x2+2x+2g(1+x) = x^2 + 2x + 2

b

x<−1x < -1

c

F(x)⋅g(x)=2x3−x2+2x−1F(x) \cdot g(x) = 2x^3 - x^2 + 2x - 1

d

F(x)−g(x)=−x2+2x−2F(x) - g(x) = -x^2 + 2x - 2

Step-by-step solution

11 steps

Here are the solutions to the problems.

: Solving for xx

Step 1: Divide both sides of the equation by 223.

223x=87223x = 87 x=87223x = \frac{87}{223}

The value of xx is approximately 0.39013.

x=87223\boxed{x = \frac{87}{223}}

: Functions
Given F(x)=2x−1F(x) = 2x - 1 and g(x)=x2+1g(x) = x^2 + 1.

Part a· 2 steps

Find g(1+x)g(1+x).

  1. Substitute (1+x)(1+x) into the expression for g(x)g(x).

    g(1+x)=(1+x)2+1g(1+x) = (1+x)^2 + 1

  2. Expand the expression.

    g(1+x)=(1+2x+x2)+1g(1+x) = (1+2x+x^2) + 1 g(1+x)=x2+2x+1+1g(1+x) = x^2 + 2x + 1 + 1 g(1+x)=x2+2x+2g(1+x) = x^2 + 2x + 2 g(1+x)=x2+2x+2\boxed{g(1+x) = x^2 + 2x + 2}

Part b

x<−1x < -1

Find the range of values of xx for which F(x)<−3F(x) < -3.

  1. Set up the inequality using the given function F(x)F(x).

    2x−1<−32x - 1 < -3

  2. Add 1 to both sides of the inequality.

    2x<−3+12x < -3 + 1 2x<−22x < -2

  3. Divide both sides by 2.

    x<−22x < \frac{-2}{2} x<−1x < -1 x<−1\boxed{x < -1}

Part c

F(x)⋅g(x)=2x3−x2+2x−1F(x) \cdot g(x) = 2x^3 - x^2 + 2x - 1

Simplify F(x)⋅g(x)F(x) \cdot g(x).

  1. Write the product of F(x)F(x) and g(x)g(x).

    F(x)⋅g(x)=(2x−1)(x2+1)F(x) \cdot g(x) = (2x - 1)(x^2 + 1)

  2. Expand the product using the distributive property.

    F(x)⋅g(x)=2x(x2)+2x(1)−1(x2)−1(1)F(x) \cdot g(x) = 2x(x^2) + 2x(1) - 1(x^2) - 1(1) F(x)⋅g(x)=2x3+2x−x2−1F(x) \cdot g(x) = 2x^3 + 2x - x^2 - 1

  3. Rearrange the terms in descending order of power.

    F(x)⋅g(x)=2x3−x2+2x−1F(x) \cdot g(x) = 2x^3 - x^2 + 2x - 1 F(x)⋅g(x)=2x3−x2+2x−1\boxed{F(x) \cdot g(x) = 2x^3 - x^2 + 2x - 1}

Part d

F(x)−g(x)=−x2+2x−2F(x) - g(x) = -x^2 + 2x - 2

Simplify F(x)−g(x)F(x) - g(x).

  1. Write the difference F(x)−g(x)F(x) - g(x).

    F(x)−g(x)=(2x−1)−(x2+1)F(x) - g(x) = (2x - 1) - (x^2 + 1)

  2. Distribute the negative sign to the terms in g(x)g(x).

    F(x)−g(x)=2x−1−x2−1F(x) - g(x) = 2x - 1 - x^2 - 1

  3. Combine like terms and arrange in descending order of power.

    F(x)−g(x)=−x2+2x−1−1F(x) - g(x) = -x^2 + 2x - 1 - 1 F(x)−g(x)=−x2+2x−2F(x) - g(x) = -x^2 + 2x - 2 F(x)−g(x)=−x2+2x−2\boxed{F(x) - g(x) = -x^2 + 2x - 2}
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