Consider the function $f(x) = \frac{e^x - 1}{e^x + 1}$. (i) Show that $f(x)$ is increasing for all $x$. (ii) Show that $f(x)$ is an odd function. (iii) Describe the behaviour of $f(x)$ for large positive values of $x$. (iv) Hence sketch the graph of $f(x) = \frac{e^x - 1}{e^x + 1}$. (v) Hence, or otherwise, sketch the graph of $y = \frac{1}{f(x)}$. Solve the quadratic equation $z^2 + (2 + 3i)z + (1 + 3i) = 0$, giving your answers in the form $a + bi$, where $a$ and $b$ are real numbers. Find $\int x \tan^{-1} x \, dx$. Let $P(x)$ be a polynomial. (i) Given that $(x - a^2)$ is a factor of $P(x)$, show that $P(a) = P(a^2) = 0$. (ii) Given that the polynomial $P(x) = x^4 - 3x^3 + x^2 + 4$ has a factor $(x - a^2)$, find the value of $a$.

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Consider the function $f(x) = \frac{e^x - 1}{e^x + 1}$ - HSC - SSCE Mathematics Extension 2 - Question 12 - 2017 - Paper 1

Question 12

$Consider-the-function-$f(x)-=-\frac{e^x---1}{e^x-+-1}$-HSC-SSCE Mathematics Extension 2-Question 12-2017-Paper 1.png$

Consider the function $f(x) = \frac{e^x - 1}{e^x + 1}$. (i) Show that $f(x)$ is increasing for all $x$. (ii) Show that $f(x)$ is an odd function. (iii) Describe t... show full transcript

Worked Solution & Example Answer:Consider the function $f(x) = \frac{e^x - 1}{e^x + 1}$ - HSC - SSCE Mathematics Extension 2 - Question 12 - 2017 - Paper 1

Step 1

Show that $f(x)$ is increasing for all $x$.

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114 rated

Answer

Step 2

Show that $f(x)$ is an odd function.

99%

104 rated

Answer

Step 3

Describe the behaviour of $f(x)$ for large positive values of $x$.

96%

101 rated

Answer

Step 4

Hence sketch the graph of $f(x) = \frac{e^x - 1}{e^x + 1}$.

98%

120 rated

Answer

Step 5

Hence, or otherwise, sketch the graph of $y = \frac{1}{f(x)}$.

97%

117 rated

Answer

Step 6

Solve the quadratic equation $z^2 + (2 + 3i)z + (1 + 3i) = 0$.

97%

121 rated

Answer

Step 7

Find $\int x \tan^{-1} x \, dx$.

96%

114 rated

Answer

Step 8

Given that $(x - a^2)$ is a factor of $P(x)$, show that $P(a) = P(a^2) = 0$.

99%

104 rated

Answer

Step 9

Given that the polynomial $P(x) = x^4 - 3x^3 + x^2 + 4$ has a factor $(x - a^2)$, find the value of $a$.

96%

101 rated

Answer

Consider the function $f(x) = \frac{e^x - 1}{e^x + 1}$ - HSC - SSCE Mathematics Extension 2 - Question 12 - 2017 - Paper 1

Worked Solution & Example Answer:Consider the function $f(x) = \frac{e^x - 1}{e^x + 1}$ - HSC - SSCE Mathematics Extension 2 - Question 12 - 2017 - Paper 1

Show that $f(x)$ is increasing for all $x$.

Show that $f(x)$ is an odd function.

Describe the behaviour of $f(x)$ for large positive values of $x$.

Hence sketch the graph of $f(x) = \frac{e^x - 1}{e^x + 1}$.

Hence, or otherwise, sketch the graph of $y = \frac{1}{f(x)}$.

Solve the quadratic equation $z^2 + (2 + 3i)z + (1 + 3i) = 0$.

Find $\int x \tan^{-1} x \, dx$.

Given that $(x - a^2)$ is a factor of $P(x)$, show that $P(a) = P(a^2) = 0$.

Given that the polynomial $P(x) = x^4 - 3x^3 + x^2 + 4$ has a factor $(x - a^2)$, find the value of $a$.

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