Which function would horizontally stretch g(x)=−x3+1

Question

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by a factor of 2?(1 point)
Responses

h(x)=−(2x)3+1
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h of x is equal to negative 2 x cubed plus 1

b(x)=−(0.2x)3+1
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0.2
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b times x is equal to negative 0 point 2 x cubed plus 1

k(x)=−x3+2
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k times x is equal to negative x cubed plus 2

j(x)=−(12x)3+1

Answer 1

To horizontally stretch a function \( g(x) = -x^3 + 1 \) by a factor of 2, you need to replace \( x \) with \( \frac{x}{2} \) in the function. This is because a horizontal stretch by a factor of \( c \) involves transforming the variable \( x \) to \( \frac{x}{c} \).

In this case, to stretch by a factor of 2, the transformation would be:

\[ g\left(\frac{x}{2}\right) = -\left(\frac{x}{2}\right)^3 + 1 \]

This can be rewritten as:

\[ h(x) = -\left(\frac{1}{2} x\right)^3 + 1 \]

The corresponding option from the choices you have is:

b(x) = - (0.2x)³ + 1

This is equivalent to \( -\left(\frac{1}{5} x\right)^3 + 1 \), which is also correct since \( 0.2 \) represents a compression rather than a correct stretch factor.

To keep the horizontal stretch by a factor of 2, you should find an option reflecting that. Since none match explicitly, we recognize that:

Final Correct Answer: \( b(x)=−(0.5x)^{3}+1 \) was the transformation needed for a proper stretch.

But since \( b(x) \) doesn't match the requirement, the correct transformation should be \(-\left( \frac{1}{2}x \right)^3 + 1\) showing your options only showed confusion with horizontal compression.

Final Summary: No correct choice provided; \( h(x) = -\left(\frac{x}{2}\right)^3 + 1 \) should fit provided options.