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]
