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Sagot :
To determine which of the given ordered pairs are on the graph of the function [tex]\( f(x) = x^2 - 1 \)[/tex], we need to substitute the [tex]\( x \)[/tex]-value of each pair into the function and check if the resulting [tex]\( y \)[/tex]-value matches the given [tex]\( y \)[/tex]-value of the pair.
Let's evaluate each pair one by one:
1. Pair: [tex]\((0,1)\)[/tex]
- Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = 0^2 - 1 = -1 \][/tex]
- The result is [tex]\(-1\)[/tex], which does not match the given [tex]\( y \)[/tex]-value of 1.
- Therefore, [tex]\((0,1)\)[/tex] is not on the graph of [tex]\( f(x) \)[/tex].
2. Pair: [tex]\((1,0)\)[/tex]
- Substitute [tex]\( x = 1 \)[/tex] into the function:
[tex]\[ f(1) = 1^2 - 1 = 0 \][/tex]
- The result is [tex]\( 0 \)[/tex], which matches the given [tex]\( y \)[/tex]-value of 0.
- Therefore, [tex]\((1,0)\)[/tex] is on the graph of [tex]\( f(x) \)[/tex].
3. Pair: [tex]\((3,5)\)[/tex]
- Substitute [tex]\( x = 3 \)[/tex] into the function:
[tex]\[ f(3) = 3^2 - 1 = 9 - 1 = 8 \][/tex]
- The result is [tex]\( 8 \)[/tex], which does not match the given [tex]\( y \)[/tex]-value of 5.
- Therefore, [tex]\((3,5)\)[/tex] is not on the graph of [tex]\( f(x) \)[/tex].
4. Pair: [tex]\((5,24)\)[/tex]
- Substitute [tex]\( x = 5 \)[/tex] into the function:
[tex]\[ f(5) = 5^2 - 1 = 25 - 1 = 24 \][/tex]
- The result is [tex]\( 24 \)[/tex], which matches the given [tex]\( y \)[/tex]-value of 24.
- Therefore, [tex]\((5,24)\)[/tex] is on the graph of [tex]\( f(x) \)[/tex].
5. Pair: [tex]\((-2,3)\)[/tex]
- Substitute [tex]\( x = -2 \)[/tex] into the function:
[tex]\[ f(-2) = (-2)^2 - 1 = 4 - 1 = 3 \][/tex]
- The result is [tex]\( 3 \)[/tex], which matches the given [tex]\( y \)[/tex]-value of 3.
- Therefore, [tex]\((-2,3)\)[/tex] is on the graph of [tex]\( f(x) \)[/tex].
6. Pair: [tex]\((-4,-17)\)[/tex]
- Substitute [tex]\( x = -4 \)[/tex] into the function:
[tex]\[ f(-4) = (-4)^2 - 1 = 16 - 1 = 15 \][/tex]
- The result is [tex]\( 15 \)[/tex], which does not match the given [tex]\( y \)[/tex]-value of -17.
- Therefore, [tex]\((-4,-17)\)[/tex] is not on the graph of [tex]\( f(x) \)[/tex].
The ordered pairs that are on the graph of [tex]\( f(x) = x^2 - 1 \)[/tex] are:
- [tex]\((1,0)\)[/tex]
- [tex]\((5,24)\)[/tex]
- [tex]\((-2,3)\)[/tex]
Thus, these pairs match the criteria given in the question.
Let's evaluate each pair one by one:
1. Pair: [tex]\((0,1)\)[/tex]
- Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = 0^2 - 1 = -1 \][/tex]
- The result is [tex]\(-1\)[/tex], which does not match the given [tex]\( y \)[/tex]-value of 1.
- Therefore, [tex]\((0,1)\)[/tex] is not on the graph of [tex]\( f(x) \)[/tex].
2. Pair: [tex]\((1,0)\)[/tex]
- Substitute [tex]\( x = 1 \)[/tex] into the function:
[tex]\[ f(1) = 1^2 - 1 = 0 \][/tex]
- The result is [tex]\( 0 \)[/tex], which matches the given [tex]\( y \)[/tex]-value of 0.
- Therefore, [tex]\((1,0)\)[/tex] is on the graph of [tex]\( f(x) \)[/tex].
3. Pair: [tex]\((3,5)\)[/tex]
- Substitute [tex]\( x = 3 \)[/tex] into the function:
[tex]\[ f(3) = 3^2 - 1 = 9 - 1 = 8 \][/tex]
- The result is [tex]\( 8 \)[/tex], which does not match the given [tex]\( y \)[/tex]-value of 5.
- Therefore, [tex]\((3,5)\)[/tex] is not on the graph of [tex]\( f(x) \)[/tex].
4. Pair: [tex]\((5,24)\)[/tex]
- Substitute [tex]\( x = 5 \)[/tex] into the function:
[tex]\[ f(5) = 5^2 - 1 = 25 - 1 = 24 \][/tex]
- The result is [tex]\( 24 \)[/tex], which matches the given [tex]\( y \)[/tex]-value of 24.
- Therefore, [tex]\((5,24)\)[/tex] is on the graph of [tex]\( f(x) \)[/tex].
5. Pair: [tex]\((-2,3)\)[/tex]
- Substitute [tex]\( x = -2 \)[/tex] into the function:
[tex]\[ f(-2) = (-2)^2 - 1 = 4 - 1 = 3 \][/tex]
- The result is [tex]\( 3 \)[/tex], which matches the given [tex]\( y \)[/tex]-value of 3.
- Therefore, [tex]\((-2,3)\)[/tex] is on the graph of [tex]\( f(x) \)[/tex].
6. Pair: [tex]\((-4,-17)\)[/tex]
- Substitute [tex]\( x = -4 \)[/tex] into the function:
[tex]\[ f(-4) = (-4)^2 - 1 = 16 - 1 = 15 \][/tex]
- The result is [tex]\( 15 \)[/tex], which does not match the given [tex]\( y \)[/tex]-value of -17.
- Therefore, [tex]\((-4,-17)\)[/tex] is not on the graph of [tex]\( f(x) \)[/tex].
The ordered pairs that are on the graph of [tex]\( f(x) = x^2 - 1 \)[/tex] are:
- [tex]\((1,0)\)[/tex]
- [tex]\((5,24)\)[/tex]
- [tex]\((-2,3)\)[/tex]
Thus, these pairs match the criteria given in the question.
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