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Sagot :
To determine which statement describes a key feature of the function [tex]\( g(x) = 2f(x) \)[/tex], we need to understand the transformation involved. The function [tex]\( g(x) \)[/tex] is created by scaling the function [tex]\( f(x) \)[/tex] vertically by a factor of 2. Here’s a detailed step-by-step solution to identify the correct statement:
1. Understanding Vertical Scaling:
- When a function [tex]\( f(x) \)[/tex] is scaled vertically by a factor of 2, it means that every y-value of [tex]\( f(x) \)[/tex] is multiplied by 2 to get [tex]\( g(x) \)[/tex]. Therefore, [tex]\( g(x) \)[/tex] is always twice the value of [tex]\( f(x) \)[/tex] at any point [tex]\( x \)[/tex].
2. Evaluating the y-intercept:
- The y-intercept of a function is the point where the graph of the function crosses the y-axis. For this point, the value of [tex]\( x \)[/tex] is 0.
- Let’s denote the y-intercept of [tex]\( f(x) \)[/tex] by the point [tex]\( (0, b) \)[/tex]. This means [tex]\( f(0) = b \)[/tex].
- Applying the transformation to find the y-intercept of [tex]\( g(x) \)[/tex]:
[tex]\[ g(0) = 2 f(0) = 2b \][/tex]
- Therefore, if [tex]\( f(x) \)[/tex] has a y-intercept at [tex]\( (0, b) \)[/tex], then [tex]\( g(x) = 2f(x) \)[/tex] will have its y-intercept at [tex]\( (0, 2b) \)[/tex].
3. Applying the Information:
- Suppose [tex]\( f(x) \)[/tex] has a y-intercept at [tex]\( (0, 1) \)[/tex]. Then, we can determine that:
[tex]\[ g(0) = 2 \times 1 = 2 \][/tex]
- Thus, [tex]\( g(x) \)[/tex] will have a y-intercept at [tex]\( (0, 2) \)[/tex].
4. Eliminating Incorrect Choices:
- A. Horizontal asymptote of [tex]\( y = -2 \)[/tex]: This is not influenced directly by the vertical scaling described.
- B. [tex]\( y \)[/tex]-intercept at [tex]\( (2,0) \)[/tex]: This is not correct, as the x-coordinate for a y-intercept must be 0.
- C. [tex]\( y \)[/tex]-intercept at [tex]\( (0, 2) \)[/tex]: This matches our derived conclusion since [tex]\( g(x) \)[/tex] scales the y-values of [tex]\( f(x) \)[/tex] by 2.
- D. Horizontal asymptote of [tex]\( y = 2 \)[/tex]: Once again, this does not directly pertain to the scaling in the problem.
Hence, the correct statement that describes a key feature of the function [tex]\( g(x) = 2 f(x) \)[/tex] is:
C. [tex]\( y \)[/tex]-intercept at [tex]\( (0, 2) \)[/tex]
1. Understanding Vertical Scaling:
- When a function [tex]\( f(x) \)[/tex] is scaled vertically by a factor of 2, it means that every y-value of [tex]\( f(x) \)[/tex] is multiplied by 2 to get [tex]\( g(x) \)[/tex]. Therefore, [tex]\( g(x) \)[/tex] is always twice the value of [tex]\( f(x) \)[/tex] at any point [tex]\( x \)[/tex].
2. Evaluating the y-intercept:
- The y-intercept of a function is the point where the graph of the function crosses the y-axis. For this point, the value of [tex]\( x \)[/tex] is 0.
- Let’s denote the y-intercept of [tex]\( f(x) \)[/tex] by the point [tex]\( (0, b) \)[/tex]. This means [tex]\( f(0) = b \)[/tex].
- Applying the transformation to find the y-intercept of [tex]\( g(x) \)[/tex]:
[tex]\[ g(0) = 2 f(0) = 2b \][/tex]
- Therefore, if [tex]\( f(x) \)[/tex] has a y-intercept at [tex]\( (0, b) \)[/tex], then [tex]\( g(x) = 2f(x) \)[/tex] will have its y-intercept at [tex]\( (0, 2b) \)[/tex].
3. Applying the Information:
- Suppose [tex]\( f(x) \)[/tex] has a y-intercept at [tex]\( (0, 1) \)[/tex]. Then, we can determine that:
[tex]\[ g(0) = 2 \times 1 = 2 \][/tex]
- Thus, [tex]\( g(x) \)[/tex] will have a y-intercept at [tex]\( (0, 2) \)[/tex].
4. Eliminating Incorrect Choices:
- A. Horizontal asymptote of [tex]\( y = -2 \)[/tex]: This is not influenced directly by the vertical scaling described.
- B. [tex]\( y \)[/tex]-intercept at [tex]\( (2,0) \)[/tex]: This is not correct, as the x-coordinate for a y-intercept must be 0.
- C. [tex]\( y \)[/tex]-intercept at [tex]\( (0, 2) \)[/tex]: This matches our derived conclusion since [tex]\( g(x) \)[/tex] scales the y-values of [tex]\( f(x) \)[/tex] by 2.
- D. Horizontal asymptote of [tex]\( y = 2 \)[/tex]: Once again, this does not directly pertain to the scaling in the problem.
Hence, the correct statement that describes a key feature of the function [tex]\( g(x) = 2 f(x) \)[/tex] is:
C. [tex]\( y \)[/tex]-intercept at [tex]\( (0, 2) \)[/tex]
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