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Let's solve the given problems step by step from the image you provided.

### d) Solve for \( x \):
1. \( 10^x > 100^{2x+5} \)

We start by rewriting \( 100 \) as \( 10^2 \):
\[
100^{2x+5} = (10^2)^{2x+5} = 10^{2(2x+5)} = 10^{4x + 10}
\]
Thus, the inequality becomes:
\[
10^x > 10^{4x + 10}
\]
Since the bases are the same, we can compare the exponents:
\[
x > 4x + 10
\]
Solving for \( x \):
\[
x - 4x > 10 \quad \Rightarrow \quad -3x > 10 \quad \Rightarrow \quad x < -\frac{10}{3}
\]

So, the solution is \( x < -\frac{10}{3} \).

---

2. \( 169^x = 13^{x+1} \)

Since \( 169 = 13^2 \), we rewrite the equation:
\[
(13^2)^x = 13^{x+1} \quad \Rightarrow \quad 13^{2x} = 13^{x+1}
\]
Since the bases are the same, we can compare the exponents:
\[
2x = x + 1
\]
Solving for \( x \):
\[
2x - x = 1 \quad \Rightarrow \quad x = 1
\]

So, the solution is \( x = 1 \).

---

### e) Graph the function \( f(x) = 2^x + 3 \)

Now, let's address the characteristics of the function:

- **Y-intercept**: This occurs when \( x = 0 \). Substituting \( x = 0 \) into the equation:
  \[
  f(0) = 2^0 + 3 = 1 + 3 = 4
  \]
  The y-intercept is \( (0, 4) \).

- **X-intercept**: To find the x-intercept, we set \( f(x) = 0 \):
  \[
  2^x + 3 = 0 \quad \Rightarrow \quad 2^x = -3
  \]
  This has no real solution, as \( 2^x \) is always positive.

- **Horizontal Asymptote (HA)**: As \( x \to -\infty \), \( 2^x \to 0 \), so the horizontal asymptote is \( y = 3 \).

- **Vertical Asymptote (VA)**: Exponential functions like \( 2^x + 3 \) do not have vertical asymptotes.

- **Domain**: The function is defined for all real numbers, so the domain is \( (-\infty, \infty) \).

- **Range**: Since the lowest value that \( 2^x \) can take is 0 (as \( x \to -\infty \)) and \( 2^x \) grows without bound as \( x \to \infty \), the range is \( (3, \infty) \).

Would you like more details or a plot for the graph? Also, here are some additional questions that might interest you:

1. What would happen if the base in the exponential equation were a fractional value?
2. How do you solve for the x-intercept of an exponential equation if one exists?
3. What transformations occur when you add constants to an exponential function?
4. How would the graph change if the function were \( f(x) = 2^{-x} + 3 \)?
5. How do logarithmic functions relate to exponential functions?

**Tip:** When solving inequalities involving exponentials, try rewriting them with the same base to compare exponents directly.

Solve for x in the equations 10^x > 100^(2x+5) and 169^x = 13^(x+1). Then, graph the function f(x) = 2^x + 3, and determine its y-intercept, x-intercept, horizontal asymptote, vertical asymptote, domain, and range.

Learn how to solve the exponential inequality 10^x > 100^(2x+5) and the equation 169^x = 13^(x+1), followed by graphing the function f(x) = 2^x + 3. This includes finding the y-intercept, x-intercept, horizontal asymptote, vertical asymptote, domain, and range.

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