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Answer :
Sure, let's break down the process of evaluating the function [tex]\( f(x) = 4x^2 - 2x - 7 \)[/tex] for different inputs step-by-step.
1. Evaluate [tex]\( f(3) \)[/tex]:
- Substitute [tex]\( x = 3 \)[/tex] into the function.
- Calculate: [tex]\( f(3) = 4(3)^2 - 2(3) - 7 \)[/tex].
- First, compute [tex]\( 4(3)^2 \)[/tex]: [tex]\( 3^2 = 9 \)[/tex], so [tex]\( 4 \times 9 = 36 \)[/tex].
- Next, compute [tex]\( -2(3) \)[/tex]: which is [tex]\(-6\)[/tex].
- Combine these: [tex]\( 36 - 6 - 7 = 23 \)[/tex].
- Therefore, [tex]\( f(3) = 23 \)[/tex].
2. Evaluate [tex]\( f(-2) \)[/tex]:
- Substitute [tex]\( x = -2 \)[/tex] into the function.
- Calculate: [tex]\( f(-2) = 4(-2)^2 - 2(-2) - 7 \)[/tex].
- First, compute [tex]\( 4(-2)^2 \)[/tex]: [tex]\( (-2)^2 = 4 \)[/tex], so [tex]\( 4 \times 4 = 16 \)[/tex].
- Next, compute [tex]\( -2(-2) \)[/tex]: which is [tex]\(4\)[/tex].
- Combine these: [tex]\( 16 + 4 - 7 = 13 \)[/tex].
- Therefore, [tex]\( f(-2) = 13 \)[/tex].
3. Evaluate [tex]\( f(a + 5) \)[/tex]:
- Substitute [tex]\( x = a + 5 \)[/tex] into the function.
- Calculate: [tex]\( f(a + 5) = 4(a + 5)^2 - 2(a + 5) - 7 \)[/tex].
- Expand [tex]\( (a + 5)^2 \)[/tex]: [tex]\( a^2 + 10a + 25 \)[/tex].
- So, [tex]\( 4(a + 5)^2 = 4(a^2 + 10a + 25) = 4a^2 + 40a + 100 \)[/tex].
- Next, compute [tex]\( -2(a + 5) = -2a - 10 \)[/tex].
- Combine these: [tex]\( 4a^2 + 40a + 100 - 2a - 10 - 7 \)[/tex].
- Simplify: [tex]\( 4a^2 + 38a + (100 - 10 - 7) \)[/tex].
- Therefore, [tex]\( f(a + 5) = 4a^2 + 38a + 83 \)[/tex].
To summarize, the results are as follows:
- [tex]\( f(3) = 23 \)[/tex]
- [tex]\( f(-2) = 13 \)[/tex]
- [tex]\( f(a + 5) = 4a^2 + 38a + 83 \)[/tex]
1. Evaluate [tex]\( f(3) \)[/tex]:
- Substitute [tex]\( x = 3 \)[/tex] into the function.
- Calculate: [tex]\( f(3) = 4(3)^2 - 2(3) - 7 \)[/tex].
- First, compute [tex]\( 4(3)^2 \)[/tex]: [tex]\( 3^2 = 9 \)[/tex], so [tex]\( 4 \times 9 = 36 \)[/tex].
- Next, compute [tex]\( -2(3) \)[/tex]: which is [tex]\(-6\)[/tex].
- Combine these: [tex]\( 36 - 6 - 7 = 23 \)[/tex].
- Therefore, [tex]\( f(3) = 23 \)[/tex].
2. Evaluate [tex]\( f(-2) \)[/tex]:
- Substitute [tex]\( x = -2 \)[/tex] into the function.
- Calculate: [tex]\( f(-2) = 4(-2)^2 - 2(-2) - 7 \)[/tex].
- First, compute [tex]\( 4(-2)^2 \)[/tex]: [tex]\( (-2)^2 = 4 \)[/tex], so [tex]\( 4 \times 4 = 16 \)[/tex].
- Next, compute [tex]\( -2(-2) \)[/tex]: which is [tex]\(4\)[/tex].
- Combine these: [tex]\( 16 + 4 - 7 = 13 \)[/tex].
- Therefore, [tex]\( f(-2) = 13 \)[/tex].
3. Evaluate [tex]\( f(a + 5) \)[/tex]:
- Substitute [tex]\( x = a + 5 \)[/tex] into the function.
- Calculate: [tex]\( f(a + 5) = 4(a + 5)^2 - 2(a + 5) - 7 \)[/tex].
- Expand [tex]\( (a + 5)^2 \)[/tex]: [tex]\( a^2 + 10a + 25 \)[/tex].
- So, [tex]\( 4(a + 5)^2 = 4(a^2 + 10a + 25) = 4a^2 + 40a + 100 \)[/tex].
- Next, compute [tex]\( -2(a + 5) = -2a - 10 \)[/tex].
- Combine these: [tex]\( 4a^2 + 40a + 100 - 2a - 10 - 7 \)[/tex].
- Simplify: [tex]\( 4a^2 + 38a + (100 - 10 - 7) \)[/tex].
- Therefore, [tex]\( f(a + 5) = 4a^2 + 38a + 83 \)[/tex].
To summarize, the results are as follows:
- [tex]\( f(3) = 23 \)[/tex]
- [tex]\( f(-2) = 13 \)[/tex]
- [tex]\( f(a + 5) = 4a^2 + 38a + 83 \)[/tex]
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