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Sagot :
Let's analyze the given functions to determine their [tex]\( y \)[/tex]-intercepts and then compare them with the [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) = |x+3| + 4 \)[/tex].
1. Calculate [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex]:
For [tex]\( g(x) = |x+3| + 4 \)[/tex], substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = |0 + 3| + 4 = 3 + 4 = 7 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is 7.
2. Calculate [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex]:
For [tex]\( f(x) = -2(x-8)^2 \)[/tex], substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -2(0-8)^2 = -2(-8)^2 = -2 \cdot 64 = -128 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is -128.
3. Calculate [tex]\( y \)[/tex]-intercept of [tex]\( h(x) \)[/tex]:
For [tex]\( h(x) = -5|x| + 10 \)[/tex], substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = -5|0| + 10 = -5 \cdot 0 + 10 = 10 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( h(x) \)[/tex] is 10.
4. Calculate [tex]\( y \)[/tex]-intercept of [tex]\( j(x) \)[/tex]:
For [tex]\( j(x) = 4(x+2)^2 + 8 \)[/tex], substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ j(0) = 4(0+2)^2 + 8 = 4 \cdot 4 + 8 = 16 + 8 = 24 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( j(x) \)[/tex] is 24.
5. Calculate [tex]\( y \)[/tex]-intercept of [tex]\( k(x) \)[/tex]:
For [tex]\( k(x) = \frac{1}{4}(x-4)^2 + 4 \)[/tex], substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ k(0) = \frac{1}{4}(0-4)^2 + 4 = \frac{1}{4} \cdot 16 + 4 = 4 + 4 = 8 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( k(x) \)[/tex] is 8.
6. Calculate [tex]\( y \)[/tex]-intercept of [tex]\( m(x) \)[/tex]:
For [tex]\( m(x) = \frac{1}{4}|x-8| + 6 \)[/tex], substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ m(0) = \frac{1}{4}|0-8| + 6 = \frac{1}{4} \cdot 8 + 6 = 2 + 6 = 8 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( m(x) \)[/tex] is 8.
Now we have the [tex]\( y \)[/tex]-intercepts for all functions:
- [tex]\( g(x) \)[/tex]: 7
- [tex]\( f(x) \)[/tex]: -128
- [tex]\( h(x) \)[/tex]: 10
- [tex]\( j(x) \)[/tex]: 24
- [tex]\( k(x) \)[/tex]: 8
- [tex]\( m(x) \)[/tex]: 8
Next, we identify which functions have a [tex]\( y \)[/tex]-intercept greater than 7 (the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex]):
- [tex]\( h(x) \)[/tex]: 10 (greater than 7)
- [tex]\( j(x) \)[/tex]: 24 (greater than 7)
- [tex]\( k(x) \)[/tex]: 8 (greater than 7)
- [tex]\( m(x) \)[/tex]: 8 (greater than 7)
Thus, the functions with a [tex]\( y \)[/tex]-intercept greater than the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) = |x+3| + 4 \)[/tex] are:
- [tex]\( h(x) = -5|x| + 10 \)[/tex]
- [tex]\( j(x) = 4(x+2)^2 + 8 \)[/tex]
- [tex]\( k(x) = \frac{1}{4}(x-4)^2 + 4 \)[/tex]
- [tex]\( m(x) = \frac{1}{4}|x-8| + 6 \)[/tex]
Among these, to focus on only three options, note that [tex]\( k(x) \)[/tex] and [tex]\( m(x) \)[/tex] both have a [tex]\( y \)[/tex]-intercept of 8. Hence, any three out of these four options are correct:
[tex]\[ h(x), j(x), k(x), \text{and}\ m(x) \text{ all have \( y \)-intercepts greater than \( g(x) \)'s \( y \)-intercept} \][/tex]
1. Calculate [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex]:
For [tex]\( g(x) = |x+3| + 4 \)[/tex], substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = |0 + 3| + 4 = 3 + 4 = 7 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is 7.
2. Calculate [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex]:
For [tex]\( f(x) = -2(x-8)^2 \)[/tex], substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -2(0-8)^2 = -2(-8)^2 = -2 \cdot 64 = -128 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is -128.
3. Calculate [tex]\( y \)[/tex]-intercept of [tex]\( h(x) \)[/tex]:
For [tex]\( h(x) = -5|x| + 10 \)[/tex], substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = -5|0| + 10 = -5 \cdot 0 + 10 = 10 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( h(x) \)[/tex] is 10.
4. Calculate [tex]\( y \)[/tex]-intercept of [tex]\( j(x) \)[/tex]:
For [tex]\( j(x) = 4(x+2)^2 + 8 \)[/tex], substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ j(0) = 4(0+2)^2 + 8 = 4 \cdot 4 + 8 = 16 + 8 = 24 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( j(x) \)[/tex] is 24.
5. Calculate [tex]\( y \)[/tex]-intercept of [tex]\( k(x) \)[/tex]:
For [tex]\( k(x) = \frac{1}{4}(x-4)^2 + 4 \)[/tex], substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ k(0) = \frac{1}{4}(0-4)^2 + 4 = \frac{1}{4} \cdot 16 + 4 = 4 + 4 = 8 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( k(x) \)[/tex] is 8.
6. Calculate [tex]\( y \)[/tex]-intercept of [tex]\( m(x) \)[/tex]:
For [tex]\( m(x) = \frac{1}{4}|x-8| + 6 \)[/tex], substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ m(0) = \frac{1}{4}|0-8| + 6 = \frac{1}{4} \cdot 8 + 6 = 2 + 6 = 8 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( m(x) \)[/tex] is 8.
Now we have the [tex]\( y \)[/tex]-intercepts for all functions:
- [tex]\( g(x) \)[/tex]: 7
- [tex]\( f(x) \)[/tex]: -128
- [tex]\( h(x) \)[/tex]: 10
- [tex]\( j(x) \)[/tex]: 24
- [tex]\( k(x) \)[/tex]: 8
- [tex]\( m(x) \)[/tex]: 8
Next, we identify which functions have a [tex]\( y \)[/tex]-intercept greater than 7 (the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex]):
- [tex]\( h(x) \)[/tex]: 10 (greater than 7)
- [tex]\( j(x) \)[/tex]: 24 (greater than 7)
- [tex]\( k(x) \)[/tex]: 8 (greater than 7)
- [tex]\( m(x) \)[/tex]: 8 (greater than 7)
Thus, the functions with a [tex]\( y \)[/tex]-intercept greater than the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) = |x+3| + 4 \)[/tex] are:
- [tex]\( h(x) = -5|x| + 10 \)[/tex]
- [tex]\( j(x) = 4(x+2)^2 + 8 \)[/tex]
- [tex]\( k(x) = \frac{1}{4}(x-4)^2 + 4 \)[/tex]
- [tex]\( m(x) = \frac{1}{4}|x-8| + 6 \)[/tex]
Among these, to focus on only three options, note that [tex]\( k(x) \)[/tex] and [tex]\( m(x) \)[/tex] both have a [tex]\( y \)[/tex]-intercept of 8. Hence, any three out of these four options are correct:
[tex]\[ h(x), j(x), k(x), \text{and}\ m(x) \text{ all have \( y \)-intercepts greater than \( g(x) \)'s \( y \)-intercept} \][/tex]
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