Answer :
Sure! Let's solve these step-by-step:
41. If [tex]\( f(x) = 2x + 1 \)[/tex] and [tex]\( g \)[/tex] is a function such that [tex]\( f(g(2)) = 3 \)[/tex], then [tex]\( g(2) \)[/tex] is equal to:
To find [tex]\( g(2) \)[/tex], we first need to substitute [tex]\( g(2) \)[/tex] into the function [tex]\( f(x) \)[/tex] and set it equal to 3.
1. The function given is [tex]\( f(x) = 2x + 1 \)[/tex].
2. We're told [tex]\( f(g(2)) = 3 \)[/tex].
3. Substitute [tex]\( g(2) \)[/tex] into the function: [tex]\( f(g(2)) = 2 \times g(2) + 1 \)[/tex].
4. Set the equation equal to 3:
[tex]\[ 2 \times g(2) + 1 = 3 \][/tex]
5. Solve for [tex]\( g(2) \)[/tex]:
[tex]\[ 2 \times g(2) = 3 - 1 \][/tex]
[tex]\[ 2 \times g(2) = 2 \][/tex]
[tex]\[ g(2) = \frac{2}{2} \][/tex]
[tex]\[ g(2) = 1 \][/tex]
So, the value of [tex]\( g(2) \)[/tex] is 1. Hence, the answer is A. 1.
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42. Let [tex]\( f(x) = 3^x \)[/tex]. Which of the following is true for each real numbers [tex]\( x \)[/tex] and [tex]\( y \)[/tex]?
We'll evaluate each option to see which one holds true:
- A. [tex]\( f(x+y) = f(x) + f(y) \)[/tex]
Substituting the function: [tex]\( 3^{x+y} \)[/tex] should equal [tex]\( 3^x + 3^y \)[/tex].
This is not true because exponent addition properties don't result in sums: [tex]\( 3^{x+y} \)[/tex] does not equal [tex]\( 3^x + 3^y \)[/tex].
- B. [tex]\( f(xy) = f(x)f(y) \)[/tex]
Check by substituting: [tex]\( 3^{xy} \)[/tex] should equal [tex]\( 3^x \times 3^y \)[/tex].
This is true since [tex]\( 3^{xy} = (3^x)^y = 3^x \times 3^y \)[/tex].
- C. [tex]\( f(x^2 + 1) = \frac{1}{3}f(x^2) \)[/tex]
Substituting, we get: [tex]\( 3^{x^2+1} \)[/tex] should equal [tex]\(\frac{1}{3} \times 3^{x^2} \)[/tex].
Simplifying the left side: [tex]\( 3^{x^2+1} = 3 \times 3^{x^2} \)[/tex] is not equal to [tex]\(\frac{1}{3} \times 3^{x^2}\)[/tex].
- D. [tex]\( f(x-1) < f(x) \)[/tex]
Substituting, we have: [tex]\( 3^{x-1} \)[/tex] should be less than [tex]\( 3^x \)[/tex].
Simplifying, [tex]\( 3^{x-1} = \frac{3^x}{3} \)[/tex], and this is indeed less than [tex]\( 3^x \)[/tex].
Options B and D are true.
Therefore, for question 42, the correct statements are B. [tex]\( f(xy) = f(x)f(y) \)[/tex] and D. [tex]\( f(x-1) < f(x) \)[/tex].
41. If [tex]\( f(x) = 2x + 1 \)[/tex] and [tex]\( g \)[/tex] is a function such that [tex]\( f(g(2)) = 3 \)[/tex], then [tex]\( g(2) \)[/tex] is equal to:
To find [tex]\( g(2) \)[/tex], we first need to substitute [tex]\( g(2) \)[/tex] into the function [tex]\( f(x) \)[/tex] and set it equal to 3.
1. The function given is [tex]\( f(x) = 2x + 1 \)[/tex].
2. We're told [tex]\( f(g(2)) = 3 \)[/tex].
3. Substitute [tex]\( g(2) \)[/tex] into the function: [tex]\( f(g(2)) = 2 \times g(2) + 1 \)[/tex].
4. Set the equation equal to 3:
[tex]\[ 2 \times g(2) + 1 = 3 \][/tex]
5. Solve for [tex]\( g(2) \)[/tex]:
[tex]\[ 2 \times g(2) = 3 - 1 \][/tex]
[tex]\[ 2 \times g(2) = 2 \][/tex]
[tex]\[ g(2) = \frac{2}{2} \][/tex]
[tex]\[ g(2) = 1 \][/tex]
So, the value of [tex]\( g(2) \)[/tex] is 1. Hence, the answer is A. 1.
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42. Let [tex]\( f(x) = 3^x \)[/tex]. Which of the following is true for each real numbers [tex]\( x \)[/tex] and [tex]\( y \)[/tex]?
We'll evaluate each option to see which one holds true:
- A. [tex]\( f(x+y) = f(x) + f(y) \)[/tex]
Substituting the function: [tex]\( 3^{x+y} \)[/tex] should equal [tex]\( 3^x + 3^y \)[/tex].
This is not true because exponent addition properties don't result in sums: [tex]\( 3^{x+y} \)[/tex] does not equal [tex]\( 3^x + 3^y \)[/tex].
- B. [tex]\( f(xy) = f(x)f(y) \)[/tex]
Check by substituting: [tex]\( 3^{xy} \)[/tex] should equal [tex]\( 3^x \times 3^y \)[/tex].
This is true since [tex]\( 3^{xy} = (3^x)^y = 3^x \times 3^y \)[/tex].
- C. [tex]\( f(x^2 + 1) = \frac{1}{3}f(x^2) \)[/tex]
Substituting, we get: [tex]\( 3^{x^2+1} \)[/tex] should equal [tex]\(\frac{1}{3} \times 3^{x^2} \)[/tex].
Simplifying the left side: [tex]\( 3^{x^2+1} = 3 \times 3^{x^2} \)[/tex] is not equal to [tex]\(\frac{1}{3} \times 3^{x^2}\)[/tex].
- D. [tex]\( f(x-1) < f(x) \)[/tex]
Substituting, we have: [tex]\( 3^{x-1} \)[/tex] should be less than [tex]\( 3^x \)[/tex].
Simplifying, [tex]\( 3^{x-1} = \frac{3^x}{3} \)[/tex], and this is indeed less than [tex]\( 3^x \)[/tex].
Options B and D are true.
Therefore, for question 42, the correct statements are B. [tex]\( f(xy) = f(x)f(y) \)[/tex] and D. [tex]\( f(x-1) < f(x) \)[/tex].
