Answer :
To solve this problem, we need to compare the given functions: [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = \left( \frac{1}{5} x \right)^2 \)[/tex].
1. Understand the function [tex]\( f(x) = x^2 \)[/tex]:
- This is a basic quadratic function with its vertex at the origin [tex]\((0,0)\)[/tex] and opens upwards.
2. Understand the function [tex]\( g(x) \)[/tex]:
- The function is given as [tex]\( g(x) = \left( \frac{1}{5} x \right)^2 \)[/tex].
3. Simplify [tex]\( g(x) \)[/tex]:
- We can rewrite [tex]\( g(x) \)[/tex] to understand its form better.
[tex]\[ g(x) = \left( \frac{1}{5} x \right)^2 = \left( \frac{1}{5} \cdot x \right)^2 = \left( \frac{1}{5} \right)^2 \cdot x^2 = \frac{1}{25} x^2 \][/tex]
4. Compare with the basic function [tex]\( f(x) \)[/tex]:
- The expression [tex]\( g(x) = \frac{1}{25} x^2 \)[/tex] implies that the quadratic function [tex]\( f(x) = x^2 \)[/tex] has been modified by multiplying the input [tex]\( x \)[/tex] by [tex]\( \frac{1}{5} \)[/tex].
5. Transformation analysis:
- When a function [tex]\( f(x) \)[/tex] is transformed to [tex]\( f\left( \frac{1}{k} x \right) \)[/tex], it involves a horizontal stretch or compression.
- Specifically, [tex]\( f\left( \frac{1}{k} x \right) \)[/tex] results in a horizontal stretch of the graph of [tex]\( f(x) \)[/tex] by a factor of [tex]\( k \)[/tex]:
- If [tex]\( k > 1 \)[/tex], the graph is horizontally stretched.
- If [tex]\( 0 < k < 1 \)[/tex], the graph is horizontally compressed.
- In our case, [tex]\( k = 5 \)[/tex] because [tex]\( \frac{1}{5} x \)[/tex] is equivalent to [tex]\( f \left( \frac{1}{5} x \right) \)[/tex], which means a stretch.
6. Conclusion:
- Therefore, the function [tex]\( g(x) = \left( \frac{1}{5} x \right)^2 \)[/tex] results in the graph of [tex]\( f(x) = x^2 \)[/tex] being horizontally stretched by a factor of 5.
Based on the above analysis, the correct multiple-choice answer is:
B. The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] horizontally stretched by a factor of 5.
1. Understand the function [tex]\( f(x) = x^2 \)[/tex]:
- This is a basic quadratic function with its vertex at the origin [tex]\((0,0)\)[/tex] and opens upwards.
2. Understand the function [tex]\( g(x) \)[/tex]:
- The function is given as [tex]\( g(x) = \left( \frac{1}{5} x \right)^2 \)[/tex].
3. Simplify [tex]\( g(x) \)[/tex]:
- We can rewrite [tex]\( g(x) \)[/tex] to understand its form better.
[tex]\[ g(x) = \left( \frac{1}{5} x \right)^2 = \left( \frac{1}{5} \cdot x \right)^2 = \left( \frac{1}{5} \right)^2 \cdot x^2 = \frac{1}{25} x^2 \][/tex]
4. Compare with the basic function [tex]\( f(x) \)[/tex]:
- The expression [tex]\( g(x) = \frac{1}{25} x^2 \)[/tex] implies that the quadratic function [tex]\( f(x) = x^2 \)[/tex] has been modified by multiplying the input [tex]\( x \)[/tex] by [tex]\( \frac{1}{5} \)[/tex].
5. Transformation analysis:
- When a function [tex]\( f(x) \)[/tex] is transformed to [tex]\( f\left( \frac{1}{k} x \right) \)[/tex], it involves a horizontal stretch or compression.
- Specifically, [tex]\( f\left( \frac{1}{k} x \right) \)[/tex] results in a horizontal stretch of the graph of [tex]\( f(x) \)[/tex] by a factor of [tex]\( k \)[/tex]:
- If [tex]\( k > 1 \)[/tex], the graph is horizontally stretched.
- If [tex]\( 0 < k < 1 \)[/tex], the graph is horizontally compressed.
- In our case, [tex]\( k = 5 \)[/tex] because [tex]\( \frac{1}{5} x \)[/tex] is equivalent to [tex]\( f \left( \frac{1}{5} x \right) \)[/tex], which means a stretch.
6. Conclusion:
- Therefore, the function [tex]\( g(x) = \left( \frac{1}{5} x \right)^2 \)[/tex] results in the graph of [tex]\( f(x) = x^2 \)[/tex] being horizontally stretched by a factor of 5.
Based on the above analysis, the correct multiple-choice answer is:
B. The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] horizontally stretched by a factor of 5.
