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Sagot :
To determine which functions have a [tex]\( y \)[/tex]-intercept greater than the [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) = |x + 3| + 4 \)[/tex], we first need to calculate the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex]. The [tex]\( y \)[/tex]-intercept is found by evaluating the function at [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = |0 + 3| + 4 = |3| + 4 = 3 + 4 = 7 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is 7.
Next, we calculate the [tex]\( y \)[/tex]-intercepts of the other given functions:
1. [tex]\( f(x) = -2(x - 8)^2 \)[/tex]
Evaluating [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -2(0 - 8)^2 = -2 \cdot 64 = -128 \][/tex]
The [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is [tex]\(-128\)[/tex].
2. [tex]\( h(x) = -5|x| + 10 \)[/tex]
Evaluating [tex]\( h(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = -5|0| + 10 = -5 \cdot 0 + 10 = 10 \][/tex]
The [tex]\( y \)[/tex]-intercept of [tex]\( h(x) \)[/tex] is [tex]\( 10 \)[/tex].
3. [tex]\( j(x) = -4(x + 2)^2 + 8 \)[/tex]
Evaluating [tex]\( j(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ j(0) = -4(0 + 2)^2 + 8 = -4 \cdot 4 + 8 = -16 + 8 = -8 \][/tex]
The [tex]\( y \)[/tex]-intercept of [tex]\( j(x) \)[/tex] is [tex]\(-8\)[/tex].
4. [tex]\( k(x) = \frac{1}{4}(x - 4)^2 + 4 \)[/tex]
Evaluating [tex]\( k(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ k(0) = \frac{1}{4}(0 - 4)^2 + 4 = \frac{1}{4} \cdot 16 + 4 = 4 + 4 = 8 \][/tex]
The [tex]\( y \)[/tex]-intercept of [tex]\( k(x) \)[/tex] is [tex]\( 8 \)[/tex].
5. [tex]\( m(x) = \frac{1}{4}|x - 8| + 6 \)[/tex]
Evaluating [tex]\( m(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ m(0) = \frac{1}{4}|0 - 8| + 6 = \frac{1}{4} \cdot 8 + 6 = 2 + 6 = 8 \][/tex]
The [tex]\( y \)[/tex]-intercept of [tex]\( m(x) \)[/tex] is [tex]\( 8 \)[/tex].
Now, we identify which of these functions have a [tex]\( y \)[/tex]-intercept greater than [tex]\( 7 \)[/tex]:
- [tex]\( f(x) = -128 \)[/tex] (not greater than 7)
- [tex]\( h(x) = 10 \)[/tex] (greater than 7)
- [tex]\( j(x) = -8 \)[/tex] (not greater than 7)
- [tex]\( k(x) = 8 \)[/tex] (greater than 7)
- [tex]\( m(x) = 8 \)[/tex] (greater than 7)
Thus, the functions with a [tex]\( y \)[/tex]-intercept greater than the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] are:
- [tex]\( h(x) \)[/tex]
- [tex]\( k(x) \)[/tex]
- [tex]\( m(x) \)[/tex]
[tex]\[ g(0) = |0 + 3| + 4 = |3| + 4 = 3 + 4 = 7 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is 7.
Next, we calculate the [tex]\( y \)[/tex]-intercepts of the other given functions:
1. [tex]\( f(x) = -2(x - 8)^2 \)[/tex]
Evaluating [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -2(0 - 8)^2 = -2 \cdot 64 = -128 \][/tex]
The [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is [tex]\(-128\)[/tex].
2. [tex]\( h(x) = -5|x| + 10 \)[/tex]
Evaluating [tex]\( h(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = -5|0| + 10 = -5 \cdot 0 + 10 = 10 \][/tex]
The [tex]\( y \)[/tex]-intercept of [tex]\( h(x) \)[/tex] is [tex]\( 10 \)[/tex].
3. [tex]\( j(x) = -4(x + 2)^2 + 8 \)[/tex]
Evaluating [tex]\( j(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ j(0) = -4(0 + 2)^2 + 8 = -4 \cdot 4 + 8 = -16 + 8 = -8 \][/tex]
The [tex]\( y \)[/tex]-intercept of [tex]\( j(x) \)[/tex] is [tex]\(-8\)[/tex].
4. [tex]\( k(x) = \frac{1}{4}(x - 4)^2 + 4 \)[/tex]
Evaluating [tex]\( k(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ k(0) = \frac{1}{4}(0 - 4)^2 + 4 = \frac{1}{4} \cdot 16 + 4 = 4 + 4 = 8 \][/tex]
The [tex]\( y \)[/tex]-intercept of [tex]\( k(x) \)[/tex] is [tex]\( 8 \)[/tex].
5. [tex]\( m(x) = \frac{1}{4}|x - 8| + 6 \)[/tex]
Evaluating [tex]\( m(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ m(0) = \frac{1}{4}|0 - 8| + 6 = \frac{1}{4} \cdot 8 + 6 = 2 + 6 = 8 \][/tex]
The [tex]\( y \)[/tex]-intercept of [tex]\( m(x) \)[/tex] is [tex]\( 8 \)[/tex].
Now, we identify which of these functions have a [tex]\( y \)[/tex]-intercept greater than [tex]\( 7 \)[/tex]:
- [tex]\( f(x) = -128 \)[/tex] (not greater than 7)
- [tex]\( h(x) = 10 \)[/tex] (greater than 7)
- [tex]\( j(x) = -8 \)[/tex] (not greater than 7)
- [tex]\( k(x) = 8 \)[/tex] (greater than 7)
- [tex]\( m(x) = 8 \)[/tex] (greater than 7)
Thus, the functions with a [tex]\( y \)[/tex]-intercept greater than the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] are:
- [tex]\( h(x) \)[/tex]
- [tex]\( k(x) \)[/tex]
- [tex]\( m(x) \)[/tex]
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