The \(\displaystyle y\)-intercept of the graph of \(\displaystyle f(x)\) is the point at which it intersects the \(\displaystyle y\)-axis. Its \(\displaystyle x\)-coordinate is 0; its \(\displaystyle y\)-coordinate is \(\displaystyle f(0)\), which can be found by substituting 0 for \(\displaystyle x\) in the definition:

\(\displaystyle f(x) = 2 \cdot 10 ^{x} - 8\)

\(\displaystyle f(0) = 2 \cdot 10 ^{0} - 8 = 2 \cdot 1 - 8 = 2 - 8 = -6\)

The \(\displaystyle y\)-intercept of the graph of \(\displaystyle f(x)\) is the point at which it intersects the \(\displaystyle y\)-axis. Its \(\displaystyle x\)-coordinate is 0; its \(\displaystyle y\)-coordinate is \(\displaystyle f(0)\), which can be found by substituting 0 for \(\displaystyle x\) in the definition:

\(\displaystyle f(x) = 2e^{x}+ 7\)

\(\displaystyle f(0) = 2e^{0}+ 7 = 2 \cdot 1 + 7 = 2+7 = 9\),

the correct choice.

The \(\displaystyle x\)-intercept(s) of the graph of \(\displaystyle f(x)\) are the point(s) at which it intersects the \(\displaystyle x\)-axis. The \(\displaystyle y\)-coordinate of each is 0,; their \(\displaystyle x\)-coordinate(s) are those value(s) of \(\displaystyle x\) for which \(\displaystyle f(x) = 0\), so set up, and solve for \(\displaystyle x\), the equation:

\(\displaystyle f(x)= 0\)

\(\displaystyle 2e^{x}+ 7 = 0\)

Subtract 7 from both sides:

\(\displaystyle 2e^{x}+ 7 - 7 = 0 - 7\)

\(\displaystyle 2e^{x}= - 7\)

Divide both sides by 2:

\(\displaystyle \frac{2e^{x}}{2}=\frac{ - 7}{2}\)

\(\displaystyle e^{x} =-\frac{ 7}{2}\)

The next step would normally be to take the natural logarithm of both sides in order to eliminate the exponent. However, the negative number \(\displaystyle -\frac{ 7}{2}\) does not have a natural logarithm. Therefore, this equation has no solution, and the graph of \(\displaystyle f(x)\) has no \(\displaystyle x\)-intercept.

The \(\displaystyle x\)-intercept(s) of the graph of \(\displaystyle f(x)\) are the point(s) at which it intersects the \(\displaystyle x\)-axis. The \(\displaystyle y\)-coordinate of each is 0,; their \(\displaystyle x\)-coordinate(s) are those value(s) of \(\displaystyle x\) for which \(\displaystyle f(x) = 0\), so set up, and solve for \(\displaystyle x\), the equation:

\(\displaystyle f(x)= 0\)

\(\displaystyle 2 \cdot 10 ^{x} - 8 = 0\)

Add 8 to both sides:

\(\displaystyle 2 \cdot 10 ^{x} - 8 + 8 = 0 + 8\)

\(\displaystyle 2 \cdot 10 ^{x} = 8\)

Divide both sides by 2:

\(\displaystyle \frac{2 \cdot 10 ^{x}}{2} =\frac{ 8}{2}\)

\(\displaystyle 10 ^{x} =4\)

Take the common logarithm of both sides to eliminate the base:

\(\displaystyle \log 10 ^{x} =\log 4\)

\(\displaystyle x=\log 4\)

Let \(\displaystyle g(x) = e^{x}\). This function is defined for any real number \(\displaystyle x\), so the domain of \(\displaystyle g(x)\) is the set of all real numbers. In terms of \(\displaystyle g(x)\),

\(\displaystyle f(x) = 2g(x-3) - 7\)

Since \(\displaystyle g(x)\) is defined for all real \(\displaystyle x\), so is \(\displaystyle g(x-3)\); it follows that \(\displaystyle f(x)\) is as well. The correct domain is the set of all real numbers.

Since a positive number raised to any power is equal to a positive number, 

\(\displaystyle e^{x-3} > 0\)

Applying the properties of inequality, we see that

\(\displaystyle 2 \cdot e^{x-3} > 2 \cdot 0\)

\(\displaystyle 2 e^{x-3} > 0\)

\(\displaystyle 2 e^{x-3} -7 > 0 - 7\)

\(\displaystyle 2 e^{x-3} -7 > - 7\)

\(\displaystyle f(x) > - 7\),

and the range of \(\displaystyle f(x)\) is the set \(\displaystyle (-7, \infty )\).