Discover a wealth of knowledge and get your questions answered at IDNLearn.com. Discover in-depth answers to your questions from our community of experienced professionals.
Sagot :
Let's break down the key features of the function [tex]\( g(x) = -f(x) + 10 \)[/tex]. We need to determine these features based on the given information about [tex]\( f(x) \)[/tex].
1. Horizontal Asymptote:
- Given that [tex]\( f(x) \)[/tex] has a horizontal asymptote at [tex]\( y = 0 \)[/tex], we translate this into the function [tex]\( g(x) = -f(x) + 10 \)[/tex]. The horizontal asymptote of [tex]\( f(x) \)[/tex] being [tex]\( y = 0 \)[/tex] implies that as [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex]. Hence, for [tex]\( g(x) \)[/tex], it transforms to [tex]\( y = -0 + 10 = 10 \)[/tex]. Therefore, the horizontal asymptote of [tex]\( g(x) \)[/tex] is [tex]\( y = 10 \)[/tex].
2. Range:
- If the range of [tex]\( f(x) \)[/tex] is [tex]\( \{ y \mid y \geq 0 \} \)[/tex], then when we apply the function transformation, [tex]\( g(x) = -f(x) + 10 \)[/tex], the minimum value of [tex]\( f(x) \)[/tex] (which is 0) will correspond to 10 in [tex]\( g(x) \)[/tex]. As [tex]\( f(x) \to \infty \)[/tex], [tex]\( -f(x) \to -\infty \)[/tex]. Hence, [tex]\( g(x) \)[/tex] will approach negative infinity. Therefore, the range of [tex]\( g(x) \)[/tex] becomes [tex]\( \{ y \mid -\infty < y < 10 \} \)[/tex].
3. Y-Intercept:
- To find the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex], set [tex]\( x = 0 \)[/tex] in [tex]\( g(x) \)[/tex]. Assume [tex]\( f(0) = 1 \)[/tex], which implies [tex]\( g(0) = -f(0) + 10 = -1 + 10 = 9 \)[/tex]. Therefore, the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is at [tex]\( (0, 9) \)[/tex].
4. X-Intercept:
- The [tex]\( x \)[/tex]-intercept occurs where [tex]\( g(x) = 0 \)[/tex]. Setting [tex]\( g(x) = 0 \)[/tex] gives [tex]\( 0 = -f(x) + 10 \)[/tex], solving for [tex]\( f(x) \)[/tex] yields [tex]\( f(x) = 10 \)[/tex]. Let's assume [tex]\( f(1) = 10 \)[/tex], then [tex]\( -f(1) + 10 = 0 \)[/tex], implying that [tex]\( g(1) = 0 \)[/tex]. Therefore, the [tex]\( x \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is at [tex]\( (1, 0) \)[/tex].
5. Increasing [tex]\( y \)[/tex]-Values:
- Given the transformation [tex]\( g(x) = -f(x) + 10 \)[/tex], if [tex]\( f(x) \)[/tex] is an increasing function, [tex]\( g(x) \)[/tex] will be a decreasing function since multiplying by [tex]\(-1\)[/tex] inverts the slope.
Based on this reasoning, the correct selections are as follows:
- Horizontal asymptote of [tex]\( y = 10 \)[/tex]
- Range of [tex]\( \{y \mid -\infty < y < 10 \} \)[/tex]
- [tex]\( y \)[/tex]-intercept at [tex]\( (0, 9) \)[/tex]
- [tex]\( x \)[/tex]-intercept at [tex]\( (1, 0) \)[/tex]
Therefore, the selected correct answers are:
1. Horizontal asymptote of [tex]\( y = 10 \)[/tex]
2. Range of [tex]\( \{y \mid -\infty < y < 10 \} \)[/tex]
3. [tex]\( y \)[/tex]-intercept at [tex]\( (0, 9) \)[/tex]
4. [tex]\( x \)[/tex]-intercept at [tex]\( (1, 0) \)[/tex]
1. Horizontal Asymptote:
- Given that [tex]\( f(x) \)[/tex] has a horizontal asymptote at [tex]\( y = 0 \)[/tex], we translate this into the function [tex]\( g(x) = -f(x) + 10 \)[/tex]. The horizontal asymptote of [tex]\( f(x) \)[/tex] being [tex]\( y = 0 \)[/tex] implies that as [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex]. Hence, for [tex]\( g(x) \)[/tex], it transforms to [tex]\( y = -0 + 10 = 10 \)[/tex]. Therefore, the horizontal asymptote of [tex]\( g(x) \)[/tex] is [tex]\( y = 10 \)[/tex].
2. Range:
- If the range of [tex]\( f(x) \)[/tex] is [tex]\( \{ y \mid y \geq 0 \} \)[/tex], then when we apply the function transformation, [tex]\( g(x) = -f(x) + 10 \)[/tex], the minimum value of [tex]\( f(x) \)[/tex] (which is 0) will correspond to 10 in [tex]\( g(x) \)[/tex]. As [tex]\( f(x) \to \infty \)[/tex], [tex]\( -f(x) \to -\infty \)[/tex]. Hence, [tex]\( g(x) \)[/tex] will approach negative infinity. Therefore, the range of [tex]\( g(x) \)[/tex] becomes [tex]\( \{ y \mid -\infty < y < 10 \} \)[/tex].
3. Y-Intercept:
- To find the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex], set [tex]\( x = 0 \)[/tex] in [tex]\( g(x) \)[/tex]. Assume [tex]\( f(0) = 1 \)[/tex], which implies [tex]\( g(0) = -f(0) + 10 = -1 + 10 = 9 \)[/tex]. Therefore, the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is at [tex]\( (0, 9) \)[/tex].
4. X-Intercept:
- The [tex]\( x \)[/tex]-intercept occurs where [tex]\( g(x) = 0 \)[/tex]. Setting [tex]\( g(x) = 0 \)[/tex] gives [tex]\( 0 = -f(x) + 10 \)[/tex], solving for [tex]\( f(x) \)[/tex] yields [tex]\( f(x) = 10 \)[/tex]. Let's assume [tex]\( f(1) = 10 \)[/tex], then [tex]\( -f(1) + 10 = 0 \)[/tex], implying that [tex]\( g(1) = 0 \)[/tex]. Therefore, the [tex]\( x \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is at [tex]\( (1, 0) \)[/tex].
5. Increasing [tex]\( y \)[/tex]-Values:
- Given the transformation [tex]\( g(x) = -f(x) + 10 \)[/tex], if [tex]\( f(x) \)[/tex] is an increasing function, [tex]\( g(x) \)[/tex] will be a decreasing function since multiplying by [tex]\(-1\)[/tex] inverts the slope.
Based on this reasoning, the correct selections are as follows:
- Horizontal asymptote of [tex]\( y = 10 \)[/tex]
- Range of [tex]\( \{y \mid -\infty < y < 10 \} \)[/tex]
- [tex]\( y \)[/tex]-intercept at [tex]\( (0, 9) \)[/tex]
- [tex]\( x \)[/tex]-intercept at [tex]\( (1, 0) \)[/tex]
Therefore, the selected correct answers are:
1. Horizontal asymptote of [tex]\( y = 10 \)[/tex]
2. Range of [tex]\( \{y \mid -\infty < y < 10 \} \)[/tex]
3. [tex]\( y \)[/tex]-intercept at [tex]\( (0, 9) \)[/tex]
4. [tex]\( x \)[/tex]-intercept at [tex]\( (1, 0) \)[/tex]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Find clear and concise answers at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.
