a) f(x) = 6x - 5 and g(x) = \frac{x + 5}{6}. Investigate if f(g(x)) = g(f(x)). b) The real variables y and x are related by y = 5x^2. (i) The equation y = 5x^2 can be rewritten in the form log_b y = a + b log_b x. Find the value of a and the value of b. (ii) Hence, or otherwise, find the real values of y for which log_5 y = 2 + log_5 \left(\frac{126}{25} x - 1\right).

Leaving Cert Mathematics Functions: a) f(x) = 6x

a) f(x) = 6x - 5 and g(x) = \frac{x + 5}{6} - Leaving Cert Mathematics - Question 3 - 2020

Question 3

$a)--f(x)-=-6x---5-and-g(x)-=-\frac{x-+-5}{6}-Leaving Cert Mathematics-Question 3-2020.png$

a) f(x) = 6x - 5 and g(x) = \frac{x + 5}{6}. Investigate if f(g(x)) = g(f(x)). b) The real variables y and x are related by y = 5x^2. (i) The equation y = 5x^2 can... show full transcript

Worked Solution & Example Answer:a) f(x) = 6x - 5 and g(x) = \frac{x + 5}{6} - Leaving Cert Mathematics - Question 3 - 2020

Step 1

Investigate if f(g(x)) = g(f(x))

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Answer

First, calculate f(g(x)):

Substitute g(x) into f: $f(g(x)) = f\left(\frac{x + 5}{6}\right) = 6\left(\frac{x + 5}{6}\right) - 5 = x + 5 - 5 = x.$

Next, calculate g(f(x)):

Substitute f(x) into g: $g(f(x)) = g(6x - 5) = \frac{(6x - 5) + 5}{6} = \frac{6x}{6} = x.$

Since both calculations yield x, we conclude:

$f(g(x)) = g(f(x)) = x.$

Step 2

Find the value of a and the value of b

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Answer

Rewrite the equation: $y = 5x^2$ Taking logs: $log_b y = log_b (5x^2)$ Using the logarithm property: $log_b y = log_b 5 + log_b (x^2) = log_b 5 + 2 log_b x.$ Thus, we have:

a = log_b 5
b = 2.

Step 3

Find the real values of y for which log_5 y = 2 + log_5 (\frac{126}{25} x - 1)

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101 rated

Answer

Starting from: $log_5 y = 2 + log_5 \left(\frac{126}{25} x - 1\right).$ Rewrite as: $log_5 y - log_5 \left(\frac{126}{25} x - 1\right) = 2.$ This leads to: $log_5 \left(\frac{y}{\frac{126}{25} x - 1}\right) = 2.$ Converting to exponential form: $\frac{y}{\frac{126}{25} x - 1} = 5^2 = 25.$ Rearranging gives: $y = 25 \left(\frac{126}{25} x - 1\right) = 126x - 25.$ This is the function relating y and x. If a quadratic form emerges, find the zeros to determine the real values.

a) f(x) = 6x - 5 and g(x) = \frac{x + 5}{6} - Leaving Cert Mathematics - Question 3 - 2020

Worked Solution & Example Answer:a) f(x) = 6x - 5 and g(x) = \frac{x + 5}{6} - Leaving Cert Mathematics - Question 3 - 2020

Investigate if f(g(x)) = g(f(x))

Find the value of a and the value of b

Find the real values of y for which log_5 y = 2 + log_5 (\frac{126}{25} x - 1)

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